Question

Prove the functions f(x) = -1/3x +4 and g(x) = 3x - 12 are inverses using composition functions.​

Answer 1

To prove that the functions f(x) = -1/3x +4 and g(x) = 3x - 12 are inverses, we need to show that their composition functions result in the identity function.

Step-by-step explanation:

To prove that the functions f(x) = -1/3x +4 and g(x) = 3x - 12 are inverses, we need to show that their composition functions result in the identity function, which means that when you apply one function after the other, you get back the original input value.

Let's start by finding the composition function (f o g)(x) = f(g(x)):

1. First, substitute g(x) into f(x). So, (f o g)(x) = f(g(x)) = f(3x - 12).
2. Next, substitute -1/3x + 4 into f(x). So, (f o g)(x) = (-1/3(3x - 12) + 4) = (-x + 4 + 4) = (-x + 8).
3. The final composition function is (-x + 8).

Now, let's find the composition function (g o f)(x) = g(f(x)):

1. First, substitute f(x) into g(x). So, (g o f)(x) = g(f(x)) = g(-1/3x + 4).
2. Next, substitute 3x - 12 into g(x). So, (g o f)(x) = (3(-1/3x + 4) - 12) = (-x + 12 - 12) = (-x).
3. The final composition function is (-x).

Since (f o g)(x) = (-x + 8) and (g o f)(x) = (-x), and both are equal to the identity function (-x), we have proven that the functions f(x) = -1/3x +4 and g(x) = 3x - 12 are inverses.