Textbook Question
Find the inverse of the function f(x)=mx, where m is a constant different from zero.
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.1.50a
a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.
Verified step by step guidance1
Recall that two functions \( h(x) \) and \( k(x) \) are inverses if and only if \( h(k(x)) = x \) and \( k(h(x)) = x \).
Start by computing the composition \( h(k(x)) \). Substitute \( k(x) = (4x)^{1/3} \) into \( h(x) = \frac{x^3}{4} \):
\[ h(k(x)) = \frac{\left((4x)^{1/3}\right)^3}{4} \]
Simplify the expression inside \( h(k(x)) \). Since raising to the power 3 and then taking the cube root are inverse operations, simplify \( \left((4x)^{1/3}\right)^3 \) to \( 4x \). Then divide by 4:
\[ h(k(x)) = \frac{4x}{4} \]
Simplify the fraction to get \( h(k(x)) = x \). Next, compute the other composition \( k(h(x)) \) by substituting \( h(x) = \frac{x^3}{4} \) into \( k(x) = (4x)^{1/3} \):
\[ k(h(x)) = \left(4 \cdot \frac{x^3}{4}\right)^{1/3} \]
Simplify inside the parentheses and then apply the cube root to verify that \( k(h(x)) = x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
Two functions are inverses if applying one function and then the other returns the original input. Formally, f and g are inverses if f(g(x)) = x and g(f(x)) = x for all x in their domains. This means each function 'undoes' the effect of the other.
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Function Composition
Function composition involves applying one function to the result of another, denoted as (f ∘ g)(x) = f(g(x)). To verify inverse functions, you compose them in both orders and check if the result simplifies to the identity function x.
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Properties of Cube and Cube Root Functions
The cube function x³ and the cube root function x^(1/3) are inverse operations because cubing and taking the cube root cancel each other out. Understanding these properties helps simplify compositions involving powers and roots.
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Related Practice
Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
43. f(x) = 5 − 4x, a = 1/2
Textbook Question
5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_3(x)
Textbook Question
Find the inverse of f(x)=-x+1. Graph the line y=-x+1 together with the line y=x. At what angle do the lines intersect?
Textbook Question
88. Given that x>0, find the maximum value, if any, of
a. x^(1/x)
Textbook Question
[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
141. a. y=arccos(cos x)