Math Stuff
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The document discusses properties and definitions related to exponents, logarithms, and exponential and logarithmic functions. It defines exponent properties like the product, quotient, power, and negative exponent properties. It also defines logarithms as the inverse of exponential functions and discusses properties of logarithms like the product, power, and change of base properties. Rational exponents and their relationship to roots is explained. The general forms of exponential and power functions are provided.
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This document discusses functions and graphs. It defines a function as a relation where each element in the domain corresponds to exactly one element in the range. Examples are provided to demonstrate determining if a relation represents a function. Function notation is introduced along with using a graphing calculator to evaluate functions. The vertical line test is described as a way to identify graphs that represent functions based on y being a function of x. Methods are provided for obtaining information from graphs like identifying the domain and range and finding intercepts.
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The document contains examples of different types of functions graphed: constant functions with the equation y=-3; absolute value functions with equations y=|x|, y=|x|+2, and y=|x-1|; a piecewise function with equations y=x-1 and y=-1; and a step function with inputs of x and outputs of 85, 170, 255, and 340. It also discusses determining the type of function a given graph represents, and provides examples of graphs representing constant, absolute value, and piecewise functions.
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The document contains examples of different types of functions graphed:
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- Absolute value functions with y=|x|, y=|x|+2, and y=|x-1|
- A step function with domain of real numbers and range of {85, 170, 255, 340}
- Examples showing the greatest integer function is a type of step function
- Examples determining if graphs represent step functions, constant functions, absolute value functions, or piecewise functions.
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This document discusses several exponent (power) laws:
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This document provides instruction on determining the domain and range of functions, identifying functions using mapping and the vertical line test, and interpreting graphs. It includes examples of determining domain and range for various functions defined by equations. Warm-up questions are provided to have students plot points and draw lines from equations. Students are reminded that their class work CW 3.4 is due for full credit.
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2. Examples are given to practice simplifying expressions, adding/subtracting roots, and rationalizing denominators. A word problem asks students to find the width of a square poster using its given area.
3. A quiz is outlined to assess understanding of simplifying roots, rationalizing denominators, and adding/subtracting roots. Students are instructed to complete worksheets and ask any remaining questions.
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Relational algebra is the basis for formal relational query languages. It uses operations like selection, projection, join, union and aggregation to query relations and return results as relations. Some key points covered are:
- Relational algebra operations include selection, projection, join, union, set differences and aggregations to query and manipulate relations.
- Relations are the operands and operations return relations, so operations can be composed to form complex queries.
- Joins combine tuples from two relations based on a join condition. Outer joins retain all tuples from one or both relations whether or not they meet the join condition.
- Aggregate operations perform functions like sum, count and average across groups of tuples in a
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.
5
Producción Escrita Matemática.
Suma, resta, multiplicación y división de expresiones algebraicas. La suma y resta de monomios y polinomios siguen las mismas reglas que las operaciones numéricas. La multiplicación requiere aplicar las propiedades de las potencias y la distribución. Existen productos notables cuya factorización se obtiene por inspección. La factorización transforma expresiones en productos de factores.
Class XI CH 2 (relations and functions)
This document provides an introduction to relations and functions. It defines ordered pairs, Cartesian products of sets, relations between sets, and different types of relations such as one-to-one and many-to-many. Functions are introduced as a special type of relation where each input has a single output. Examples of real-valued functions are given such as identity, constant, modulus, and greatest integer functions. Algebraic operations on real functions like addition, subtraction, multiplication by a scalar, and multiplication of two functions are also described. The document concludes with some practice questions.
Chpt 2-functions-seqs v.5
1. The document discusses various mathematical functions including absolute value, floor/ceiling functions, factorials, modular arithmetic, exponential functions, logarithmic functions, and polynomials.
2. It defines key properties of functions such as one-to-one, onto, bijective, and inverse functions. It also covers function composition and important sequences like geometric and arithmetic progressions.
3. The document provides formulas for summing sequences and common summations like the sum of the first n natural numbers and the sum of the first n odd integers.
Exponential and logarithmic functions
1) The document discusses graphing and properties of exponential and logarithmic functions, including: graphing exponential functions by substituting values of the variable into the equation, graphing logarithmic functions using the change of base formula, and properties like the product, quotient, and power properties of logarithms.
2) Examples are provided of solving exponential and logarithmic equations using properties like changing bases to the same value, multiplying or dividing arguments using the product and quotient properties, and applying exponents using the power property.
3) Steps shown include using properties to isolate the variable, set arguments or exponents equal to each other, and solve the resulting equation.
BP106RMT.pdf
This document provides information about the key concepts and formulas related to differential calculus. It defines the derivative as the rate of change of a function with respect to an independent variable. Some common derivative formulas are presented for various standard functions like polynomial, exponential, logarithmic and trigonometric functions. Examples are given to demonstrate calculating derivatives using the formulas. The concepts of maxima and minima of functions are also introduced along with an example to find the maximum and minimum values of a given function without using derivatives.
Logarithms
This document discusses properties and laws of logarithms. It contains the following key points:
1. Logarithms represent exponents - the logarithm of a number is the exponent that the base must be raised to to produce that number.
2. Basic properties of logarithms include: logarithms are only defined for positive values; log 1 = 0 and log b = 1 for any base b; and the property that b^log(x) = x.
3. The product and quotient laws state that for logarithms with the same base, log(ab) = log(a) + log(b) and log(a/b) = log(a) - log(b).
function
1) Functions relate inputs to outputs through ordered pairs where each input maps to exactly one output. The domain is the set of inputs and the range is the set of outputs.
2) There are different types of functions including linear, quadratic, and composition functions. A linear function's graph is a straight line while a quadratic function's graph is a parabola.
3) Composition functions combine other functions where the output of one becomes the input of another. Together functions provide a powerful modeling tool used across many fields including medicine.
Lesson 4- Math 10 - W4Q1_General Form of a Polynomial Function.pptx
The document discusses polynomial functions. It defines polynomials as algebraic expressions consisting of terms in the form axn where a is a real number and n is a non-negative integer. The degree of a polynomial refers to the highest exponent of its terms. A polynomial function P(x) is written as P(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0 where n is the degree. The document provides examples of finding the degree of polynomials and performing operations like addition and multiplication on polynomial functions. It also demonstrates using synthetic division to divide one polynomial by another.
Mathematics - Functions.pdf
This document provides information about Baraka Loibanguti, who is the author of an advanced mathematics book. It includes his contact information and some notes about copyright and permissions. The document then begins discussing functions, including definitions of domain, range, and different types of functions like linear, quadratic, cubic, and polynomial functions. It provides examples of how to graph different types of functions by creating tables of values or using intercepts.
Composition and inverse of functions
The document discusses composition of functions and inverse functions. It defines composition of functions as combining two functions where one function is performed first and the result is substituted into the second function. The composition is not always commutative. It then provides examples of finding the composition of two functions. Inverse functions are defined as functions where the independent and dependent variables are swapped, and their composition is equal to x. The document demonstrates finding the inverse of functions by swapping variables and checking the composition. It emphasizes that the inverse of a function may not always be a function itself.
Pre-Cal 40S April 17, 2009
This document discusses logarithmic and exponential functions. It defines logarithms as exponents and notes that logarithmic and exponential functions are inverses of each other. It provides examples of using properties like the product, quotient, and power laws to simplify logarithmic expressions.
Presentation2.pptx
Evaluating a function means replacing a variable in the function with a value from the domain and computing the result. For example, to evaluate f(x) = 2x + 4 for x = 5, replace x with 5: f(5) = 2×5 + 4 = 14. It is important to see that the variable is just a placeholder and the name of the function is arbitrary. Evaluating a function can mean replacing the variable with a number or an expression.
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Lesson 4- Math 10 - W4Q1_General Form of a Polynomial Function.pptx
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- 1. 5.2 – Exponents and Power Functions Product Property- a^m * a^n = a^m+n Quotient Property- a^m/a^n = a^m-n Definiton of Negative- a^-n = 1/a^n Zero Exponents- a^0 = 1 Power of a Power- (a^m)^n = a^m*n Power of a Product- (a*b)^m = a^m*b^m Power Property of Equality- if a = b then a^n = b^n General form of an exponential Function is y = a*b^x General form of a power function is Y = a*x^n
- 2. 5.3 – Rational Exponents and Roots - Properties of exponents only give one solution to an equation because they are only defined for positive bases. - Point-Ratio form is y = y1 * b^x-x1 - x^1/5 is the same as the fifth root of x. EX. 9 th root of b^5 th = 26 B^5/9 = 26 (b^5/9)^9/5 = 26^9/5 B = 352.33
- 3. 5.6 – Logarithmic Functions -log x is another way of expressing x as a power of 10, which is the commonly used base for logarithms. -a logarithm is: for a > 0 and b > 0, (logb)a = x is equivalent to a = b^x -Log change of base = (logb)a = loga/logb EX: 4^x = 128 4^3 = 64 and 4^4 = 256, so must be between 3 and 4 (10log4)^x = (10log128) 10^x log 4 = 10log128 X log 4 = log 128 X = log 128/log 4 = 3.5
- 4. 5.7- Properties of Logarithms -Definition of Logarithm: if x = a^m, then (loga)x = m -Product Property: a^m * a^n = a^m+n -Quotient Property: a^m/a^n = a^m-n -Power Property: (loga)x^n = n (loga)x -Power of a Power Property: (a^m)^n = a^m*n -Power of a Quotient Property: (a/b)^n = a^n/b^n -Change of Base: (loga)x = (logb)x/(logb)a -Def. Of Rational Exponents: a^m/n = n root of a ^ m -Def. Of Negative Exponents: a^-n = 1/a^n