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 and g(x) = -x

Verified step by step guidance

1. Identify the given functions: \(f(x) = -x\) and \(g(x) = -x\).
2. Find the composition \(f(g(x))\) by substituting \(g(x)\) into \(f\): write \(f(g(x)) = f(-x)\).
3. Evaluate \(f(-x)\) by replacing the input of \(f\) with \(-x\): since \(f(t) = -t\), then \(f(-x) = -(-x)\).
4. Simplify \(f(g(x))\): \(-(-x) = x\).
5. Similarly, find \(g(f(x))\) by substituting \(f(x)\) into \(g\): write \(g(f(x)) = g(-x)\), then evaluate \(g(-x) = -(-x) = x\). Since both compositions equal \(x\), conclude that \(f\) and \(g\) are inverses of each other.

Key Concepts

Function Composition

Function composition involves applying one function to the result of another, denoted as f(g(x)). It means substituting g(x) into f(x), which helps analyze how two functions interact and combine their effects.

Inverse Functions

Inverse functions reverse each other's operations, so f(g(x)) = x and g(f(x)) = x for all x in the domain. Identifying inverses requires checking if composing the functions in both orders returns the original input.

Properties of Linear Functions

Linear functions have the form f(x) = mx + b. Understanding their behavior, especially when m = -1 and b = 0 as in f(x) = -x, is essential for evaluating compositions and determining if two linear functions are inverses.