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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 95
Chapter 3, Problem 95

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If f(x)=3xf(x) = 3x and g(x)=x+5g(x) = x + 5, find (f∘g)−1(x)\(\left\)(f\(\circ\) g\(\right\))^{-1}(x) and (g−1∘f−1)(x)(g^{-1} \(\circ\) f^{-1})(x).

Verified step by step guidance
1
First, understand the notation: (ƒ 0 g)(x) means the composition of functions f and g, which is f(g(x)). Similarly, (ƒ 0 g)^{-1}(x) means the inverse of the composition f(g(x)).
Step 1: Find the composition (ƒ 0 g)(x) by substituting g(x) into f. Since f(x) = 3x and g(x) = x + 5, write the expression for f(g(x)) as \(f(g(x)) = 3(x + 5)\).
Step 2: To find the inverse of the composition, set \(y = 3(x + 5)\) and solve for x in terms of y. This involves isolating x on one side of the equation.
Step 3: Next, find the inverse functions individually: find \(f^{-1}(x)\) by solving \(y = 3x\) for x, and find \(g^{-1}(x)\) by solving \(y = x + 5\) for x.
Step 4: Finally, find the composition \((g^{-1} 0 f^{-1})(x)\) by substituting \(f^{-1}(x)\) into \(g^{-1}\). Write the expression for \(g^{-1}(f^{-1}(x))\).

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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 applying one function to the result of another, denoted as (f ∘ g)(x) = f(g(x)). Understanding how to combine functions correctly is essential for evaluating expressions like (f ∘ g)(x) and manipulating them for further operations.
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Function Composition

Inverse Functions

An inverse function reverses the effect of the original function, such that f(f⁻¹(x)) = x. Finding the inverse requires solving for x in terms of y and swapping variables. Recognizing and computing inverses is crucial for expressions involving f⁻¹ or g⁻¹.
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Graphing Logarithmic Functions

Notation and Order of Operations in Compositions and Inverses

Understanding the notation (f ∘ g)⁻¹ and (g⁻¹ ∘ f⁻¹) requires knowing that the inverse of a composition reverses the order: (f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹. This concept helps correctly interpret and simplify composite inverse functions.
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Multiplying & Dividing Functions
Related Practice
Textbook Question

Let f and g be defined by the following table: Find f(−1)−f(0)−[g(2)]2+f(−2)÷g(2)⋅g(−1).\(\sqrt{f(-1)-f(0)}\)-[g(2)]^2+f(-2)\(\div\) g(2)\(\cdot\) g(-1).

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Textbook Question

Begin by graphing the absolute value function, f(x) = |x|. Then use transformations of this graph to graph the given function. g(x) = -2|x+3|+2

Textbook Question

The functions in Exercises 93–95 are all one-to-one. For each function, (a) find an equation for f^(-1)x, the inverse function. (b) Verify that your equation is correct by showing that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. f(x) = (x - 7)/(x + 2)

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Textbook Question

Let f(x) = x² − x + 4 and g(x) = 3x – 5. Find g(-1) and f(g(-1)).

Textbook Question

Let f(x) = x² − x + 4 and g(x) = 3x – 5. Find g (1) and f(g(1)).

Textbook Question

Find all values of x satisfying the given conditions. f(x) = 2x − 5, g(x) = x² − 3x + 8, and (ƒ o g) (x) = 7.