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) = 4x and g(x) = x/4

Verified step by step guidance

1. First, find the composition \( f(g(x)) \). This means you substitute \( g(x) \) into \( f(x) \). Since \( f(x) = 4x \) and \( g(x) = \frac{x}{4} \), write \( f(g(x)) = f\left( \frac{x}{4} \right) \).
2. Next, evaluate \( f\left( \frac{x}{4} \right) \) by replacing the \( x \) in \( f(x) = 4x \) with \( \frac{x}{4} \). This gives \( f\left( \frac{x}{4} \right) = 4 \times \frac{x}{4} \).
3. Simplify the expression \( 4 \times \frac{x}{4} \) by canceling the 4 in numerator and denominator, resulting in \( f(g(x)) = x \).
4. Now, find the composition \( g(f(x)) \). Substitute \( f(x) \) into \( g(x) \). Since \( g(x) = \frac{x}{4} \) and \( f(x) = 4x \), write \( g(f(x)) = g(4x) \).
5. Evaluate \( g(4x) \) by replacing \( x \) in \( g(x) = \frac{x}{4} \) with \( 4x \), giving \( g(4x) = \frac{4x}{4} \). Simplify this to get \( g(f(x)) = x \). Since both compositions \( f(g(x)) \) and \( g(f(x)) \) equal \( x \), the functions \( f \) 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)) or g(f(x)). It means substituting the entire function g(x) into f(x), or vice versa, to create a new function. Understanding this helps evaluate combined functions and analyze their behavior.

Inverse Functions

Inverse functions reverse the effect of each other, so applying one after the other returns the original input: f(g(x)) = x and g(f(x)) = x. To determine if two functions are inverses, their compositions must simplify to the identity function, meaning the output equals the input.

Linear Functions and Their Properties

Linear functions have the form f(x) = mx + b, where m and b are constants. In this question, both functions are linear with no constant term, making it easier to analyze their compositions and inverses. Recognizing linearity simplifies calculations and understanding of function behavior.