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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 1.96a
Chapter 1, Problem 1.96a

Inverse of composite functions


a. Let g(x) = 2x + 3 and h(x) = x³. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

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Start by finding the composite function \( f(x) = g(h(x)) \). Substitute \( h(x) = x^3 \) into \( g(x) = 2x + 3 \) to get \( f(x) = 2(x^3) + 3 = 2x^3 + 3 \).
To find the inverse \( f^{-1}(x) \), set \( y = f(x) = 2x^3 + 3 \) and solve for \( x \) in terms of \( y \).
Rearrange the equation \( y = 2x^3 + 3 \) to isolate \( x^3 \): \( y - 3 = 2x^3 \).
Divide both sides by 2 to get \( x^3 = \frac{y - 3}{2} \).
Take the cube root of both sides to solve for \( x \): \( x = \sqrt[3]{\frac{y - 3}{2}} \). This gives the inverse function \( f^{-1}(x) = \sqrt[3]{\frac{x - 3}{2}} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Composite Functions

A composite function is formed when one function is applied to the result of another function. In this case, if we have functions g(x) and h(x), the composite function f(x) = g(h(x)) means we first apply h to x and then apply g to the result of h. Understanding how to manipulate and evaluate composite functions is crucial for finding their inverses.
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Evaluate Composite Functions - Special Cases

Inverse Functions

An inverse function essentially reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f⁻¹(y) takes y back to x. To find the inverse of a composite function, we often need to find the inverses of the individual functions involved and apply them in reverse order.
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Inverse Cosine

Function Notation and Operations

Understanding function notation and operations is essential for working with functions and their inverses. This includes knowing how to denote functions, apply them, and manipulate their expressions. In the context of the question, recognizing how to express the inverse of a composite function in terms of the inverses of its components (g⁻¹ and h⁻¹) is key to solving the problem.
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Multiplying & Dividing Functions
Related Practice
Textbook Question

Splitting up curves The unit circle x² + y² = 1  consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x)  (see figure) <IMAGE>.


a. Find the domain and a formula for each function.

Textbook Question

Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.

f(x) = 7 / (x + 3)

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Textbook Question

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f-1(1 + f(-3))

Textbook Question

Inverse of composite functions


b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Textbook Question

Solving equations Solve each equation.


ln 3x + ln (x + 2) = 0

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


If y= 3ˣ , then x = ³√y

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