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

Verified step by step guidance

1. Recall that two functions \( h(x) \) and \( k(x) \) are inverses if and only if \( h(k(x)) = x \) and \( k(h(x)) = x \).
2. Start by computing the composition \( h(k(x)) \). Substitute \( k(x) = (4x)^{1/3} \) into \( h(x) = \frac{x^3}{4} \): \[ h(k(x)) = \frac{\left((4x)^{1/3}\right)^3}{4} \]
3. 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:
4. \[ h(k(x)) = \frac{4x}{4} \]
5. 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 \( k(h(x)) = x \).

Key Concepts

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.