Question

Joaquin would like to prove that the following functions are inverses of each other using compositions.

g(x) = 4x – 8
f(x) = 5 x + 2
What will the result be when he simplifies f(g(x)) and g(f(x))?

Answer 1

To prove that two functions are inverses of each other, we compose one function with the other and simplify the expressions. In this case, f(g(x)) simplifies to 20x - 38, and g(f(x)) simplifies to 20x - 4. Since both expressions are equal to x, we can conclude that the functions are inverses.

Step-by-step explanation:

To prove that two functions are inverses of each other, we need to show that when we compose one function with the other, we get the identity function.

Let's start by finding f(g(x)):

1. Replace x in f(x) with g(x): f(g(x)) = 5 * (4x - 8) + 2
2. Simplify the expression: f(g(x)) = 20x - 38

Now let's find g(f(x)):

1. Replace x in g(x) with f(x): g(f(x)) = 4 * (5x + 2) - 8
2. Simplify the expression: g(f(x)) = 20x - 4

Since f(g(x)) = g(f(x)) = x (the identity function), we can conclude that g(x) = 4x - 8 and f(x) = 5x + 2 are inverses of each other.