Use the definition of inverses to determine whether ƒ and g are inverses.
$f\left(x\right)=-4x+2,g\left(x\right)=-\frac{1}{4}x-2$

Verified step by step guidance

1. Recall that two functions \( f \) and \( g \) are inverses if and only if \( f(g(x)) = x \) and \( g(f(x)) = x \) for all \( x \) in the domains of the compositions.
2. Start by finding the composition \( f(g(x)) \). Substitute \( g(x) = -\frac{1}{4}x - 2 \) into \( f(x) = -4x + 2 \), so \( f(g(x)) = -4 \left(-\frac{1}{4}x - 2\right) + 2 \).
3. Simplify the expression for \( f(g(x)) \) by distributing \( -4 \) and combining like terms carefully.
4. Next, find the composition \( g(f(x)) \). Substitute \( f(x) = -4x + 2 \) into \( g(x) = -\frac{1}{4}x - 2 \), so \( g(f(x)) = -\frac{1}{4}(-4x + 2) - 2 \).
5. Simplify the expression for \( g(f(x)) \) by distributing \( -\frac{1}{4} \) and combining like terms, then check if both compositions simplify to \( x \). If they do, \( f \) and \( g \) are inverses.

Key Concepts

Definition of Inverse Functions

Two functions f and g are inverses if applying one after the other returns the original input, meaning f(g(x)) = x and g(f(x)) = x for all x in their domains. This relationship shows that each function 'undoes' the effect of the other.

Function Composition

Function composition involves substituting one function into another, denoted as (f ∘ g)(x) = f(g(x)). To verify inverses, you compute both f(g(x)) and g(f(x)) and check if both simplify to x, confirming the inverse relationship.

Linear Functions and Their Inverses

Linear functions have the form f(x) = mx + b, where m ≠ 0. Their inverses are also linear and can be found by solving for x in terms of y. Understanding how to manipulate and invert linear functions is essential to verify if two given linear functions are inverses.