The document discusses inverse relations and inverse functions. It defines an inverse relation as interchanging the x and y variables of a relation. An inverse function exists when the inverse relation is also a function, as determined by the horizontal line test or uniqueness of solutions. Examples are provided to illustrate inverse relations that are functions and those that are not. Properties of inverse functions are defined, including that applying the function and its inverse returns the original value.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
Unit 4 lesson 3 remediation activity 1ncvpsmanage3
A function is a special type of relation where each input is mapped to exactly one output. For a relation to be a function, the inputs or x-values cannot repeat. The document provides examples of relations and identifies which ones are functions based on this criterion. It also explains that the vertical line test can be used to determine if a graphed relation is a function - if a vertical line intersects the graph more than once, it is not a function.
11 X1 T02 06 relations and functions (2010)Nigel Simmons
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document discusses inverse functions and relations. It defines an inverse relation as one where the coordinates of a relation are switched, and an inverse function as one where the domain and range of a function are switched. It provides examples of finding the inverse of specific relations and functions by switching their coordinates or domain and range. It also discusses how to determine if two functions are inverses using their graphs and the horizontal line test.
The document discusses odd and even functions. Odd functions satisfy f(-x) = -f(x) and have point symmetry about the origin. Examples given include y = x3 and y = x7 - x5. Even functions satisfy f(-x) = f(x) and have line symmetry about the y-axis. Examples provided are y = x2 and y = x2 + 4. The key characteristics of odd and even functions are outlined and examples are used to prove that certain functions, such as y = x3 + x7, are odd functions.
The document discusses differentiability and implicit differentiation. It defines differentiability as when the left and right-sided limits of the derivative at a point are equal, making the curve smooth. It provides examples of functions that are and aren't differentiable at certain points. Implicit differentiation is introduced as taking the derivative of both sides of an equation simultaneously to find derivatives that aren't explicit functions. Examples demonstrate finding derivatives of implicit functions using this method.
Fungsi distribusi adalah fungsi F yang memenuhi 3 sifat: (i) F(-∞) = 0 dan F(∞) = 1, (ii) F tidak turun, dan (iii) F kontinu dari kanan. Fungsi distribusi mendefinisikan variabel acak X dengan F(x) = P{w; Xw ≤ x} untuk semua x, sehingga F mewakili probabilitas X ≤ x. Fungsi distribusi memenuhi sifat-sifat fungsi distribusi berdasarkan definisi variabel acak.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
Unit 4 lesson 3 remediation activity 1ncvpsmanage3
A function is a special type of relation where each input is mapped to exactly one output. For a relation to be a function, the inputs or x-values cannot repeat. The document provides examples of relations and identifies which ones are functions based on this criterion. It also explains that the vertical line test can be used to determine if a graphed relation is a function - if a vertical line intersects the graph more than once, it is not a function.
11 X1 T02 06 relations and functions (2010)Nigel Simmons
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document discusses inverse functions and relations. It defines an inverse relation as one where the coordinates of a relation are switched, and an inverse function as one where the domain and range of a function are switched. It provides examples of finding the inverse of specific relations and functions by switching their coordinates or domain and range. It also discusses how to determine if two functions are inverses using their graphs and the horizontal line test.
The document discusses odd and even functions. Odd functions satisfy f(-x) = -f(x) and have point symmetry about the origin. Examples given include y = x3 and y = x7 - x5. Even functions satisfy f(-x) = f(x) and have line symmetry about the y-axis. Examples provided are y = x2 and y = x2 + 4. The key characteristics of odd and even functions are outlined and examples are used to prove that certain functions, such as y = x3 + x7, are odd functions.
The document discusses differentiability and implicit differentiation. It defines differentiability as when the left and right-sided limits of the derivative at a point are equal, making the curve smooth. It provides examples of functions that are and aren't differentiable at certain points. Implicit differentiation is introduced as taking the derivative of both sides of an equation simultaneously to find derivatives that aren't explicit functions. Examples demonstrate finding derivatives of implicit functions using this method.
Fungsi distribusi adalah fungsi F yang memenuhi 3 sifat: (i) F(-∞) = 0 dan F(∞) = 1, (ii) F tidak turun, dan (iii) F kontinu dari kanan. Fungsi distribusi mendefinisikan variabel acak X dengan F(x) = P{w; Xw ≤ x} untuk semua x, sehingga F mewakili probabilitas X ≤ x. Fungsi distribusi memenuhi sifat-sifat fungsi distribusi berdasarkan definisi variabel acak.
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
The document discusses relationships between the coefficients and roots of polynomials. It shows that:
- The sum of the roots equals the coefficient of the x term divided by the leading coefficient
- The sum of the products of the roots taken two at a time equals the coefficient of the x^2 term divided by the leading coefficient
- These relationships can be generalized to higher order terms, relating the sums of roots taken n at a time to the coefficients.
11 x1 t05 04 point slope formula (2012)Nigel Simmons
The document discusses the point slope formula and provides examples of its use. Specifically, it:
1) Defines the point slope formula as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point.
2) Works through an example of finding the equation of the line passing through points (-3,4) and (2,-6).
3) Works through another example of finding the equation of the line passing through (2,-3) that is parallel to the given equation 3x + 4y - 5 = 0.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
11 x1 t02 04 rationalising the denominator (2012)Nigel Simmons
The document discusses rationalizing denominators through four examples:
1) (i)^2 is rationalized to 4/2 = 2
2) (ii)^3/2 is rationalized to 3√5/√5
3) (iii)/(2-1) is rationalized to 3√2+3/(2-1)
4) (iv)/(2-3) is rationalized to (2+3)(2+3)(2+3)/(4-3)
The document discusses properties of transversals and parallel lines. It shows examples of parallel lines cut by a transversal and defines the ratio of intercepts theorem - that the ratio of corresponding intercepts of parallel lines cut by a transversal is equal. An example problem demonstrates using the ratio of intercepts theorem to find the length of an unknown intercept given the other intercept lengths.
The document discusses motion around a banked curve. It defines the horizontal and vertical forces involved when an object moves around such a curve. It shows that for no sideways force, the horizontal centrifugal force must equal the horizontal component of the normal force from the bank. This allows deriving an equation that relates the angle of the bank, radius of the curve, and required speed to eliminate sideways forces. As an example, it calculates the most favorable speed for a train moving around a curved track with a given radius and bank angle.
The document discusses representing complex numbers geometrically using vectors on an Argand diagram. It states that complex numbers can be represented as vectors, with the advantage that vectors can be moved around the diagram while preserving their length and angle. It describes how to add and subtract complex numbers by placing the vectors head to tail or head to head, forming a parallelogram. It also discusses the triangular inequality property as it applies to complex number addition and subtraction.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. Inverse functions satisfy the properties f^-1(f(x)) = x and f(f^-1(x)) = x. Examples are provided to demonstrate testing if an inverse function exists.
1. An inverse relation maps the outputs of a function back to the inputs by switching the domain and range.
2. To find the inverse of a function, switch x and y and solve for y.
3. Two functions are inverse functions if applying one function after the other returns the original input.
1) An inverse function f^-1 interchanges the x and y values of the original function f.
2) For a function to have an inverse, it must be one-to-one, meaning each x maps to a single y and vice versa.
3) To find the inverse of a function algebraically, interchange x and y and solve for y. The graph of an inverse is the reflection of the original graph across the line y=x.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The document discusses inverse functions. It defines inverse functions as pairs of one-to-one functions that undo each other. To find the inverse of a function, interchange the x and y variables and solve for y. The graph of an inverse function is a reflection of the original function across the line y = x. Examples are provided to illustrate how to determine if a function is one-to-one, find the inverse function, and relate the graph of a function to its inverse.
This document discusses inverse functions. It explains that an inverse function switches the x and y values of the original function. It provides examples of finding the inverse of functions like y=2x-4 and verifying two functions are inverses. It also discusses graphing the inverses of quadratic and cubic functions and whether they pass the horizontal line test to be considered a function. The document ends with assigning homework problems related to inverse functions.
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
- Finding inverses of functions and identifying even and odd functions.
- Exponential and logarithmic functions including their properties and rules.
- Trigonometric and hyperbolic functions.
This document discusses one-to-one functions and inverse functions. It provides examples of functions and their inverses, and methods for finding the inverse of a function both graphically and algebraically. A key point is that a function must be one-to-one to have an inverse function, while non one-to-one relations only have an inverse relation. The inverse of a function "undoes" the original function. Functions and their inverses are reflections over the line y = x.
The document discusses inverse functions. It defines an inverse function as switching the x's and y's in a function's ordered pairs. If this inversion results in another function, it is called the inverse function. The domain of the original function becomes the range of the inverse function, and vice versa. It provides examples of finding the expressions for inverse functions and checking if they are correct inverses. It also discusses the differences between one-to-one and many-to-one functions, and uses the horizontal line test to determine which is being graphed. Exercises are provided to find specific inverse functions.
The document discusses relationships between the coefficients and roots of polynomials. It shows that:
- The sum of the roots equals the coefficient of the x term divided by the leading coefficient
- The sum of the products of the roots taken two at a time equals the coefficient of the x^2 term divided by the leading coefficient
- These relationships can be generalized to higher order terms, relating the sums of roots taken n at a time to the coefficients.
11 x1 t05 04 point slope formula (2012)Nigel Simmons
The document discusses the point slope formula and provides examples of its use. Specifically, it:
1) Defines the point slope formula as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point.
2) Works through an example of finding the equation of the line passing through points (-3,4) and (2,-6).
3) Works through another example of finding the equation of the line passing through (2,-3) that is parallel to the given equation 3x + 4y - 5 = 0.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
11 x1 t02 04 rationalising the denominator (2012)Nigel Simmons
The document discusses rationalizing denominators through four examples:
1) (i)^2 is rationalized to 4/2 = 2
2) (ii)^3/2 is rationalized to 3√5/√5
3) (iii)/(2-1) is rationalized to 3√2+3/(2-1)
4) (iv)/(2-3) is rationalized to (2+3)(2+3)(2+3)/(4-3)
The document discusses properties of transversals and parallel lines. It shows examples of parallel lines cut by a transversal and defines the ratio of intercepts theorem - that the ratio of corresponding intercepts of parallel lines cut by a transversal is equal. An example problem demonstrates using the ratio of intercepts theorem to find the length of an unknown intercept given the other intercept lengths.
The document discusses motion around a banked curve. It defines the horizontal and vertical forces involved when an object moves around such a curve. It shows that for no sideways force, the horizontal centrifugal force must equal the horizontal component of the normal force from the bank. This allows deriving an equation that relates the angle of the bank, radius of the curve, and required speed to eliminate sideways forces. As an example, it calculates the most favorable speed for a train moving around a curved track with a given radius and bank angle.
The document discusses representing complex numbers geometrically using vectors on an Argand diagram. It states that complex numbers can be represented as vectors, with the advantage that vectors can be moved around the diagram while preserving their length and angle. It describes how to add and subtract complex numbers by placing the vectors head to tail or head to head, forming a parallelogram. It also discusses the triangular inequality property as it applies to complex number addition and subtraction.
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. To test if an inverse function exists, use the horizontal line test or check if rewriting the inverse relation as y=g(x) yields a unique expression for y. If an inverse function exists, f^-1(f(x)) = x and f(
The document discusses inverse functions. An inverse function f^-1(x) is obtained by interchanging x and y in the original function f(x). For f^-1(x) to be a function, there must be a unique y-value for each x-value. A function and its inverse are reflections across the line y=x. The domain of f(x) is the range of f^-1(x), and vice versa. Inverse functions satisfy the properties f^-1(f(x)) = x and f(f^-1(x)) = x. Examples are provided to demonstrate testing if an inverse function exists.
1. An inverse relation maps the outputs of a function back to the inputs by switching the domain and range.
2. To find the inverse of a function, switch x and y and solve for y.
3. Two functions are inverse functions if applying one function after the other returns the original input.
1) An inverse function f^-1 interchanges the x and y values of the original function f.
2) For a function to have an inverse, it must be one-to-one, meaning each x maps to a single y and vice versa.
3) To find the inverse of a function algebraically, interchange x and y and solve for y. The graph of an inverse is the reflection of the original graph across the line y=x.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The document discusses inverse functions. It defines inverse functions as pairs of one-to-one functions that undo each other. To find the inverse of a function, interchange the x and y variables and solve for y. The graph of an inverse function is a reflection of the original function across the line y = x. Examples are provided to illustrate how to determine if a function is one-to-one, find the inverse function, and relate the graph of a function to its inverse.
This document discusses inverse functions. It explains that an inverse function switches the x and y values of the original function. It provides examples of finding the inverse of functions like y=2x-4 and verifying two functions are inverses. It also discusses graphing the inverses of quadratic and cubic functions and whether they pass the horizontal line test to be considered a function. The document ends with assigning homework problems related to inverse functions.
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
- Finding inverses of functions and identifying even and odd functions.
- Exponential and logarithmic functions including their properties and rules.
- Trigonometric and hyperbolic functions.
This document discusses one-to-one functions and inverse functions. It provides examples of functions and their inverses, and methods for finding the inverse of a function both graphically and algebraically. A key point is that a function must be one-to-one to have an inverse function, while non one-to-one relations only have an inverse relation. The inverse of a function "undoes" the original function. Functions and their inverses are reflections over the line y = x.
The document discusses inverse functions. It defines an inverse function as switching the x's and y's in a function's ordered pairs. If this inversion results in another function, it is called the inverse function. The domain of the original function becomes the range of the inverse function, and vice versa. It provides examples of finding the expressions for inverse functions and checking if they are correct inverses. It also discusses the differences between one-to-one and many-to-one functions, and uses the horizontal line test to determine which is being graphed. Exercises are provided to find specific inverse functions.
This document discusses inverse functions and their derivatives. It defines inverse functions as switching the x- and y-values of a function to "undo" the original function. A function has an inverse only if it passes the horizontal line test. The derivative of an inverse function at a point equals the reciprocal of the derivative of the original function at the corresponding point.
The document discusses inverse relations and functions. An inverse relation interchanges the input and output values of the original relation and swaps the domain and range. The graph of an inverse function is a reflection of the original relation across the line y=x. A function has an inverse function if and only if its graph passes the horizontal line test, meaning no horizontal line intersects the graph more than once.
The document discusses inverse functions. It defines an inverse function as reversing the components of ordered pairs in a function. Inverse functions have a special relationship where they can "undo" the changes made by the original function. A function has an inverse function if it is one-to-one, meaning each input is mapped to a single unique output. The inverse of a function can be found by switching the x and y variables and solving for y. Graphs of a function and its inverse are reflections of each other across the line y = x.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input value to at most one output value. The domain of a function is the set of all possible input values, while looking for values the input could not be to determine the domain. Examples are provided of determining domains for various functions, such as ensuring denominators are not equal to zero and that roots are not taken of negative numbers.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
The document discusses inverse functions and how an inverse function undoes the operations of the original function. It provides examples of finding the inverse of functions by switching the x and y values and solving for y. The inverse of a function will be a function itself only if the original function passes the horizontal line test.
The document discusses functions and their derivatives. It defines functions, different types of functions, and notation used for functions. It then covers the concept of limits, theorems on limits, and limits at infinity. The document defines the slope of a tangent line to a curve and increments. It provides definitions and rules for derivatives, including differentiation from first principles and various differentiation rules. It includes examples of finding derivatives using these rules and taking multiple derivatives.
Similar to 11 x1 t02 08 inverse functions (2012) (20)
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
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4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
2. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y x 3 x inverse relation is x y 3 y
3. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y x 3 x inverse relation is x y 3 y
The domain of the relation is the range of its inverse relation
4. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y x 3 x inverse relation is x y 3 y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
5. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y x 3 x inverse relation is x y 3 y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the line y = x.
6. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y x 3 x inverse relation is x y 3 y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the line y = x.
e.g. y x 2
7. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y x 3 x inverse relation is x y 3 y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the line y = x.
e.g. y x 2
domain: all real x
range: y 0
8. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y x 3 x inverse relation is x y 3 y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the line y = x.
e.g. y x 2 inverse relation: x y 2
domain: all real x
range: y 0
9. Inverse Relations
If y = f(x) is a relation, then the inverse relation obtained by
interchanging x and y is x = f(y)
e.g. y x 3 x inverse relation is x y 3 y
The domain of the relation is the range of its inverse relation
The range of the relation is the domain of its inverse relation
A relation and its inverse relation are reflections of each other in
the line y = x.
e.g. y x 2 inverse relation: x y 2
domain: all real x domain: x 0
range: y 0 range: all real y
11. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
12. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test
13. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
14. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y
x
15. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y
x
16. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y
x
Only has an
inverse relation
17. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y
x
Only has an OR
inverse relation x y2
y x
18. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y
x
Only has an OR
inverse relationx y2
y x
NOT UNIQUE
19. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y ii y x 3 y
x x
Only has an OR
inverse relationx y2
y x
NOT UNIQUE
20. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y ii y x 3 y
x x
Only has an OR
inverse relationx y2
y x
NOT UNIQUE
21. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y ii y x 3 y
x x
Only has an OR
Has an
inverse relationx y2
inverse function
y x
NOT UNIQUE
22. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y ii y x 3 y
x x
Only has an OR OR
Has an
inverse relationx y2 x y3
inverse function
y x y3 x
NOT UNIQUE
23. Inverse Functions
If an inverse relation of a function, is a function, then it is called
an inverse function.
Testing For Inverse Functions
(1) Use a horizontal line test OR
2 When x f y is rewritten as y g x , y g x is unique.
i y x 2 y ii y x 3 y
x x
Only has an OR OR
Has an
inverse relationx y2 x y3
inverse function
y x y3 x
NOT UNIQUE UNIQUE
24. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
25. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. x2
f x
x2
26. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. x2
f x
x2
x2 y2
y x
x2 y2
27. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. x2
f x
x2
x2 y2
y x
x2 y2
y 2 x y 2
xy 2 x y 2
x 1 y 2 x 2
2x 2
y
1 x
28. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. x2 x22
f x 2
x2 x 2
f 1 f x
x2
x2 y2 1
y x x 2
x2 y2
y 2 x y 2
xy 2 x y 2
x 1 y 2 x 2
2x 2
y
1 x
29. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. x2 x22
f x 2
x2 x 2
f 1 f x
x2
x2 y2 1
y x x 2
x2 y2
2x 4 2x 4
y 2 x y 2
x2 x2
xy 2 x y 2
4x
x 1 y 2 x 2
4
2x 2 x
y
1 x
30. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. x2 x22 2x 2 2
f x 2
x2 x 2 1 x
f 1 f x f f 1 x
x2 2x 2 2
x2 y2 1
y x x 2 1 x
x2 y2
2x 4 2x 4
y 2 x y 2
x2 x2
xy 2 x y 2
4x
x 1 y 2 x 2
4
2x 2 x
y
1 x
31. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an
inverse function), then;
f 1 f x x AND f f 1 x x
e.g. x2 x22 2x 2 2
f x 2
x2 x 2 1 x
f 1 f x f f 1 x
x2 2x 2 2
x2 y2 1
y x x 2 1 x
x2 y2
2x 4 2x 4 2x 2 2 2x
y 2 x y 2
x2 x2 2x 2 2 2x
xy 2 x y 2
4x 4x
x 1 y 2 x 2
4 4
2x 2 x x
y
1 x