a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.

Verified step by step guidance

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.

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.

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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