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
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 79a
Express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x).
h(x) = |2x-5|
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with expressing the given function h(x) = |2x - 5| as a composition of two functions f(x) and g(x), such that h(x) = (f ∘ g)(x). This means h(x) = f(g(x)).
Step 2: Identify the inner function g(x). The expression inside the absolute value, 2x - 5, can be treated as the inner function. Let g(x) = 2x - 5.
Step 3: Identify the outer function f(x). The absolute value operation is applied to the result of g(x). Therefore, the outer function is f(x) = |x|.
Step 4: Verify the composition. Substitute g(x) into f(x) to ensure the composition matches h(x). f(g(x)) = f(2x - 5) = |2x - 5|, which is the original function h(x).
Step 5: Conclude that the functions are f(x) = |x| and g(x) = 2x - 5, and their composition satisfies h(x) = (f ∘ g)(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Composition
Function composition involves combining two functions, where the output of one function becomes the input of another. If we have two functions f(x) and g(x), the composition is denoted as (f o g)(x) = f(g(x)). Understanding this concept is crucial for expressing the function h(x) as a composition of two simpler functions.
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Function Composition
Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function is essential in the given problem because h(x) = |2x - 5| requires us to consider how to express the linear transformation 2x - 5 in a way that can be composed with another function to yield the absolute value.
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Linear Functions
A linear function is a polynomial function of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In the context of the problem, recognizing that 2x - 5 is a linear function helps in identifying suitable functions f and g that can be composed to achieve the desired absolute value output.
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Related Practice
Textbook Question
Find the domain of each function.
Textbook Question
Begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = 2√(x+1)-1
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Textbook Question
In Exercises 77–92, use the graph to determine a. the function's range; b. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
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 function's domain; b. the function's range; c. the x-intercepts, if any; d. the y-intercept, if any; and e. the missing function values, indicated by question marks, below each graph.
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