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

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the compositions f(g(x)) and g(f(x)) for the given functions f(x) = 4x and g(x) = x/4. Then, determine if these functions are inverses of each other by checking if f(g(x)) = x and g(f(x)) = x.
2. Step 2: Compute f(g(x)). Substitute g(x) = x/4 into f(x) = 4x. This means replacing x in f(x) with g(x). The expression becomes f(g(x)) = 4 * (x/4). Simplify this expression to check if it equals x.
3. Step 3: Compute g(f(x)). Substitute f(x) = 4x into g(x) = x/4. This means replacing x in g(x) with f(x). The expression becomes g(f(x)) = (4x)/4. Simplify this expression to check if it equals x.
4. Step 4: Verify if f(g(x)) = x and g(f(x)) = x. If both conditions are true, then f(x) and g(x) are inverses of each other. If either condition fails, they are not inverses.
5. Step 5: Conclude whether f(x) and g(x) are inverses based on the results of the simplifications in Steps 2 and 3. If both compositions simplify to x, then the functions are inverses; otherwise, they are not.

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. For functions f and g, the composition f(g(x)) means applying g first and then f to the result. This concept is essential for evaluating the expressions in the question.

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other. If f and g are inverses, then f(g(x)) = x and g(f(x)) = x for all x in their domains. Understanding how to determine if two functions are inverses is crucial for solving the problem presented.

Linear Functions

Linear functions are polynomial functions of degree one, represented in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In this question, both f(x) = 4x and g(x) = x/4 are linear functions, which simplifies the process of finding their compositions and checking for inverses.