Textbook Question
Express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x).
h(x) = |2x-5|
Ch. 2 - Functions and Graphs
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 3, Problem 79
Find the domain of each function.
Verified step by step guidance1
Identify the function given: \(f(x) = \frac{x}{x^2 + 4x - 21}\).
Recall that the domain of a function includes all real numbers except where the denominator is zero, because division by zero is undefined.
Set the denominator equal to zero to find the values to exclude: \(x^2 + 4x - 21 = 0\).
Factor the quadratic equation: find two numbers that multiply to \(-21\) and add to \(4\). This gives \((x + 7)(x - 3) = 0\).
Solve each factor for zero: \(x + 7 = 0\) gives \(x = -7\), and \(x - 3 = 0\) gives \(x = 3\). These values are excluded from the domain.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain excludes values that make the denominator zero, as division by zero is undefined.
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Domain Restrictions of Composed Functions
Factoring Quadratic Expressions
Factoring involves rewriting a quadratic expression as a product of two binomials. This helps identify the roots or zeros of the quadratic, which are critical for determining values that make the denominator zero in rational functions.
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06:08Solving Quadratic Equations by Factoring
Solving Quadratic Equations
Solving quadratic equations means finding the values of x that satisfy the equation when set equal to zero. These solutions indicate points where the denominator of a rational function is zero and must be excluded from the domain.
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06:08Solving Quadratic Equations by Factoring
Related Practice
Textbook Question
Find the value of y if the line through the two given points is to have the indicated slope. (3, y) and (1, 4), m = −3
Textbook Question
In Exercises 77–92, use the graph to determine a. the function's domain; b. the function's range; and e. the missing function values, indicated by question marks, below each graph.
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views
Textbook Question
Use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
Textbook Question
In Exercises 77–92, use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
2
views
Textbook Question
Use the graph to determine a.the x-intercepts, if any; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.