Question

How are these inverse functions of each other by showing the f^-1(f(x)) = x and f(f^-1(x)) = x. If f(x) = 3/(x - 4) and f^-1(x) = (3/x) + 4? Show work.

A. The functions are not inverses of each other.
B. The functions are inverses of each other.
C. The inverse function is undefined.
D. Inverse functions cannot be determined.

Answer 1

To show that two functions are inverses of each other, we need to verify two conditions: f^-1(f(x)) = x and f(f^-1(x)) = x. By substituting the given functions into these conditions and simplifying, we can show that the functions f(x) = 3/(x - 4) and f^-1(x) = (3/x) + 4 are inverses of each other.

Step-by-step explanation:

To show that two functions are inverses of each other, we need to verify two conditions: f^-1(f(x)) = x and f(f^-1(x)) = x.

Let's start with the first condition: f^-1(f(x)) = x.

Substituting f(x) = 3/(x - 4) into f^-1(x), we get f^-1(f(x)) = f^-1(3/(x - 4)).

Using the formula for f^-1(x) = (3/x) + 4, we substitute the value of x. Thus, f^-1(3/(x - 4)) = (3/(3/(x-4))) + 4.

Simplifying further, f^-1(f(x)) = (3/(3/(x-4))) + 4 = (x-4) + 4 = x - 4 + 4 = x.

Therefore, the first condition is satisfied.

Now, let's verify the second condition: f(f^-1(x)) = x.

Substituting f^-1(x) = (3/x) + 4 into f(x), we get f(f^-1(x)) = f((3/x) + 4).

Using the formula for f(x) = 3/(x - 4), we substitute the value of x. Thus, f((3/x) + 4) = 3/((3/x) + 4 - 4).

Simplifying further, f(f^-1(x)) = 3/((3/x) + 4 - 4) = 3/(3/x) = 3x/3 = x.

Therefore, both conditions are satisfied and the functions f(x) = 3/(x - 4) and f^-1(x) = (3/x) + 4 are inverses of each other.

Hence, the correct answer is B. The functions are inverses of each other.