Day 4 examples u1f13
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This document discusses inverses of functions and their graphs. It provides an example of finding the inverse of a point (-4, 3) by reflecting it over the line y=x and switching the x and y values. It also gives an example function f(x)=4x+3 and explains how to write the inverse equation, draw the graph, and state the domain and range of both the function and its inverse. Finally, it provides two functions and explains how to determine algebraically if they are inverses of each other.
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Day 5 examples u1w14
The document discusses inverses of functions and their graphs. It states that the inverse of a function is found by reflecting the graph over the line y=x, which means switching the x and y values of each point. It gives the example of finding the inverse of the point (-4, 3), and finding the inverse of the function f(x) = 4x + 3 by switching x and y. It asks the reader to write the equation for the inverse of f(x) = 4x + 3, and to state the domain and range of each function and its inverse.
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This document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are discussed, including that they have a horizontal asymptote of y = 0 and increase or decrease exponentially as x increases or decreases. Examples are given of sketching the graphs of specific exponential functions, including stating their domains and ranges. The special exponential function f(x) = ex is also introduced, where e is the irrational number approximately equal to 2.718.
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This document introduces function notation and how to evaluate functions. It explains that f(x) represents the output of the function f for a given input x. Examples show different functions defined using this notation, such as f(x) = 2x + 1. The document also demonstrates how to evaluate functions for given x-values, find x-values for which a function is a certain amount, and graph functions using this notation. It compares translating and shifting graphs of functions.
7.7 one to_one_functions_-_inverse_functions
A function f is one-to-one if each element in its domain maps to a unique element in its range. A function is not one-to-one if two or more elements in the domain map to the same element in the range. The inverse of a one-to-one function f is the function f-1 that undoes the mapping of f by interchanging the domain and range and solving for the input given the output.
Inverse functions
The document defines inverse functions and provides examples. An inverse function f-1(x) undoes the original function f(x) so that f-1(f(x)) = x. For a function to have an inverse, it must be one-to-one meaning each output of f(x) corresponds to only one input x. The document gives examples of linear functions that are invertible and the function y=x2 that is not invertible because it is not one-to-one. It also states that if a function f(x) is one-to-one on its domain, then it has an inverse function and the domain of f(x) is equal to the range of the
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The document provides instructions for graphing functions with reciprocals. It outlines six steps: 1) Find and sketch any vertical asymptotes where the denominator is zero. 2) Find and sketch any horizontal asymptotes based on the degrees of the numerator and denominator. 3) Find and plot the y-intercept by evaluating f(0). 4) Find the x-intercept by solving the numerator. 5) Use sign analysis to determine where the function is positive and negative. 6) Use smooth curves to complete the graph.
Exponential functions
The document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are that they increase rapidly as x increases and have a horizontal asymptote of y = 0 if a > 1 or y = 0 if 0 < a < 1. Examples are given of sketching graphs of exponential functions and stating their domains and ranges. The graph of the natural exponential function f(x) = ex is also discussed.
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The document discusses inverses of functions and their graphs. It states that the inverse of a function is found by reflecting the graph over the line y=x, which means switching the x and y values of each point. It gives the example of finding the inverse of the point (-4, 3), and finding the inverse of the function f(x) = 4x + 3 by switching x and y. It asks the reader to write the equation for the inverse of f(x) = 4x + 3, and to state the domain and range of each function and its inverse.
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This document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are discussed, including that they have a horizontal asymptote of y = 0 and increase or decrease exponentially as x increases or decreases. Examples are given of sketching the graphs of specific exponential functions, including stating their domains and ranges. The special exponential function f(x) = ex is also introduced, where e is the irrational number approximately equal to 2.718.
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This document introduces function notation and how to evaluate functions. It explains that f(x) represents the output of the function f for a given input x. Examples show different functions defined using this notation, such as f(x) = 2x + 1. The document also demonstrates how to evaluate functions for given x-values, find x-values for which a function is a certain amount, and graph functions using this notation. It compares translating and shifting graphs of functions.
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A function f is one-to-one if each element in its domain maps to a unique element in its range. A function is not one-to-one if two or more elements in the domain map to the same element in the range. The inverse of a one-to-one function f is the function f-1 that undoes the mapping of f by interchanging the domain and range and solving for the input given the output.
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The document defines inverse functions and provides examples. An inverse function f-1(x) undoes the original function f(x) so that f-1(f(x)) = x. For a function to have an inverse, it must be one-to-one meaning each output of f(x) corresponds to only one input x. The document gives examples of linear functions that are invertible and the function y=x2 that is not invertible because it is not one-to-one. It also states that if a function f(x) is one-to-one on its domain, then it has an inverse function and the domain of f(x) is equal to the range of the
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The document provides instructions for graphing functions with reciprocals. It outlines six steps: 1) Find and sketch any vertical asymptotes where the denominator is zero. 2) Find and sketch any horizontal asymptotes based on the degrees of the numerator and denominator. 3) Find and plot the y-intercept by evaluating f(0). 4) Find the x-intercept by solving the numerator. 5) Use sign analysis to determine where the function is positive and negative. 6) Use smooth curves to complete the graph.
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The document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are that they increase rapidly as x increases and have a horizontal asymptote of y = 0 if a > 1 or y = 0 if 0 < a < 1. Examples are given of sketching graphs of exponential functions and stating their domains and ranges. The graph of the natural exponential function f(x) = ex is also discussed.
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This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.
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Identify the end behavior of a function
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1) The document discusses finding the inverse of functions by interchanging the x and y variables and solving for y. It provides examples of finding the inverses of f(x)=3x-7 and g(x)=2x^3+1.
2) It also discusses verifying inverses by checking if the composition of a function and its inverse equals x. And finding inverses of functions with restricted domains, including an example of f(x)=sqrt(x+4).
3) Finally, it discusses the relationship between a function being one-to-one and having an inverse function, both algebraically and graphically.
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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.
Addition and Subtraction of Functions
The document discusses finding the sum of two functions f(x) and g(x). It provides two examples:
1) f(x)=x^2 and g(x)=2. The sum function h(x)=x^2+2 is graphed by adding the y-values.
2) f(x)=x^2 and g(x)=3x. The sum function p(x)=x^2+3x is graphed by adding the y-values. The domain is all real numbers and the range is all real numbers greater than -9/4, as determined by finding the midpoint of the function.
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- Graphing piecewise functions
- Finding inverses and compositions of functions
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The questions cover skills like solving, graphing, and analyzing properties of different types of functions.
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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.
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The document discusses various concepts related to functions and graphs:
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2) It explains the terminology used in functions such as domain, co-domain, pre-image, and image.
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The document discusses inverse functions, logarithmic functions, and their properties. It defines an inverse function f^-1(x) as satisfying f(f^-1(x)) = x. It also defines the logarithm log_a(x) as the inverse of the exponential function a^x. Key properties of inverse functions and logarithms are outlined, including: the derivative of an inverse function using the inverse function theorem; logarithm rules such as log_a(xy) = log_a(x) + log_a(y); and converting between logarithmic bases using ln(x)/ln(a). Examples of evaluating and graphing inverse functions and logarithms are provided.
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1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
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This document covers exponential and logarithmic functions, including their definitions, graphs, properties, and applications. Some key points are:
- Exponential functions have the form f(x) = bx where b is the base. Their graphs stretch upwards and have asymptotes at the x-axis.
- The natural exponential function is f(x) = ex.
- Logarithmic functions are the inverses of exponential functions and have the form f(x) = logbx. Their graphs stretch downwards and have asymptotes at the y-axis.
- Properties of exponents and logarithms include the change of base formula and rules for simplifying expressions.
- Exponential and logarithmic equations can be solved
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Solve an exponential equation using base 10 logarithms.
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The document discusses polynomial functions and their properties. It defines turning points as points where the graph of a function changes from increasing to decreasing. It then categorizes common polynomial functions as constant, linear, quadratic, cubic, or quartic depending on their degree. It provides examples of each type of polynomial function and notes that the maximum number of zeros is equal to the degree of the polynomial function.
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This document discusses how to graph polynomial functions and find local extrema. It provides instructions on making a table of values, plotting points, connecting them with a smooth curve, and checking end behavior based on the degree and leading coefficient. Extrema are defined as local maxima or minima, where the graph changes from increasing to decreasing. Examples are given to demonstrate graphing polynomials and finding turning points that indicate local extrema.
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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.
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1) The document discusses finding the inverse of functions by interchanging the x and y variables and solving for y. It provides examples of finding the inverses of f(x)=3x-7 and g(x)=2x^3+1.
2) It also discusses verifying inverses by checking if the composition of a function and its inverse equals x. And finding inverses of functions with restricted domains, including an example of f(x)=sqrt(x+4).
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Composite functions refer to combining two functions, where the output of one function is used as the input of the other. The notation used is (f ◦ g)(x) = f(g(x)), where the inner function g(x) is evaluated first, and then substituted into the outer function f(x). Some examples are given of evaluating composite functions using function tables or graphs.
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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.
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The document discusses finding the sum of two functions f(x) and g(x). It provides two examples:
1) f(x)=x^2 and g(x)=2. The sum function h(x)=x^2+2 is graphed by adding the y-values.
2) f(x)=x^2 and g(x)=3x. The sum function p(x)=x^2+3x is graphed by adding the y-values. The domain is all real numbers and the range is all real numbers greater than -9/4, as determined by finding the midpoint of the function.
Review
The document contains a chapter review with multiple choice questions covering various algebra topics including:
- Using the quadratic formula to solve quadratic equations
- Solving linear and quadratic equations
- Finding the domain and range of functions
- Identifying intervals of increase/decrease for functions
- Graphing piecewise functions
- Finding inverses and compositions of functions
- Shifting graphs of standard functions
The questions cover skills like solving, graphing, and analyzing properties of different types of functions.
4.1 Inverse 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.
Functions and graphs
The document discusses various concepts related to functions and graphs:
1) It defines what a function is and provides examples of different types of functions such as identity, constant, polynomial, and rational functions.
2) It explains the terminology used in functions such as domain, co-domain, pre-image, and image.
3) It discusses the properties of one-to-one, onto, and bijective functions and provides examples of each. The concepts of inverse and composition of functions are also introduced.
4) Useful mathematical functions like floor, ceiling, and round are defined. Concepts related to limits, continuity, differentiability of functions and their graphs are explained briefly.
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Lesson 15: Inverse Functions And Logarithms
The document discusses inverse functions, logarithmic functions, and their properties. It defines an inverse function f^-1(x) as satisfying f(f^-1(x)) = x. It also defines the logarithm log_a(x) as the inverse of the exponential function a^x. Key properties of inverse functions and logarithms are outlined, including: the derivative of an inverse function using the inverse function theorem; logarithm rules such as log_a(xy) = log_a(x) + log_a(y); and converting between logarithmic bases using ln(x)/ln(a). Examples of evaluating and graphing inverse functions and logarithms are provided.
Day 5 examples u6w14
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
Math12 lesson11
This document covers exponential and logarithmic functions, including their definitions, graphs, properties, and applications. Some key points are:
- Exponential functions have the form f(x) = bx where b is the base. Their graphs stretch upwards and have asymptotes at the x-axis.
- The natural exponential function is f(x) = ex.
- Logarithmic functions are the inverses of exponential functions and have the form f(x) = logbx. Their graphs stretch downwards and have asymptotes at the y-axis.
- Properties of exponents and logarithms include the change of base formula and rules for simplifying expressions.
- Exponential and logarithmic equations can be solved
4.3 Logarithmic Functions
Solve an exponential equation using base 10 logarithms.
Find the missing part of a logarithm equation.
Use properties of logarithms to solve problems.
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Day 4 examples u7w14
This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.
Day 3 examples u7w14
Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.
Day 2 examples u7w14
Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
Day 1 examples u7w14
This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.
Day 7 examples u6w14
This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.
Day 4 examples u6w14
The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.
Day 3 examples u6w14
Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Day 2 examples u6w14
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
Day 1 examples u6w14
This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.
Mental math test outline
The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.
Day 8 examples u5w14
The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.
Day 7 examples u5w14
The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.
Day 5 examples u5w14
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
Day 3 examples u5w14
This document asks which of the following binomials are factors of the expression 9x - 12. The options given are x + 3 and x - 2. Of these two options, x - 2 is a factor of 9x - 12, since (3x - 4)(3x - 4) = 9x^2 - 12x + 12 = 9x - 12.
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Day 4 examples u1f13
- 2. Inverses of Functions + their Graphs ex. Find the inverse of (-4 , 3) Inverse = the reflection of the function over the line y = x which we can get by switching x with y for all points ex. f(x) = 4x + 3. Draw f
- 3. Given f(x) = 4x + 3, write an equation for f
- 5. State the domain and range of each function and its inverse.
- 6. Determine algebraically whether the functions are inverses of each other: y = (x - 2)2 + 1, x ≤ 2 y = √x - 1 + 2