Textbook Question
31. The incidence of a disease (Continuation of Example 4.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.
a. How long will it take to reduce the number of cases to 1000?
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- 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
Find the inverse of the function f(x)=mx, where m is a constant different from zero.
Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
43. f(x) = 5 − 4x, a = 1/2
Textbook Question
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
a. 10x^4 + 30x + 1
1
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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)