In Exercises 1-10, find f(g(x)) and g (f(x)) and determine whether each pair of functions ƒ and g are inverses of each other. f(x) = ∛(x − 4) and g(x) = x³ +4

Verified step by step guidance

1. Step 1: To find \( f(g(x)) \), substitute \( g(x) = x^3 + 4 \) into \( f(x) = \sqrt[3]{x - 4} \). This gives \( f(g(x)) = \sqrt[3]{(x^3 + 4) - 4} \).
2. Step 2: Simplify the expression \( f(g(x)) = \sqrt[3]{x^3} \).
3. Step 3: To find \( g(f(x)) \), substitute \( f(x) = \sqrt[3]{x - 4} \) into \( g(x) = x^3 + 4 \). This gives \( g(f(x)) = (\sqrt[3]{x - 4})^3 + 4 \).
4. Step 4: Simplify the expression \( g(f(x)) = (x - 4) + 4 \).
5. Step 5: Determine if \( f(g(x)) = x \) and \( g(f(x)) = x \). If both are true, then \( f \) and \( g \) are inverses of each other.

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. For example, if f(x) and g(x) are two functions, the composition f(g(x)) means you first apply g to x, then apply f to the result. Understanding this concept is crucial for solving the problem as it requires calculating both f(g(x)) and g(f(x)).

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If f(x) is a function, its inverse, denoted as f⁻¹(x), satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f. To determine if f and g are inverses, one must check if f(g(x)) = x and g(f(x)) = x, which is essential for the problem at hand.

Cube Root and Cubing

The cube root function, represented as ∛(x), is the inverse operation of cubing a number (x³). This relationship is fundamental in the given functions f(x) and g(x), as f(x) involves taking the cube root, while g(x) involves cubing. Recognizing how these operations interact is key to evaluating the compositions and verifying if the functions are inverses.