Question

Which of the following pairs of functions are inverses of each other?

a. f(x) = 10 and g(x) = 9(x + 7) - 10
b. f(x) = 9x - 8 and g(x) = 2(x - 9)
c. f(x) = 8(x - 3) + 4 and g(x) = 3x + 4
d. f(x) = x^2 + 5 and g(x) = x^2 - 5

Answer 1

The pair of functions (c) f(x) = 8(x - 3) + 4 and g(x) = 3x + 4 are inverses of each other.

Step-by-step explanation:

The pair of functions (c) f(x) = 8(x - 3) + 4 and g(x) = 3x + 4 are inverses of each other.

To prove this, we need to show that f(g(x)) = x and g(f(x)) = x for all values of x.

Let's evaluate f(g(x)) = f(3x + 4).

Substitute 3x + 4 into f(x) and simplify:

f(g(x)) = 8((3x + 4) - 3) + 4 = 8(3x + 1) + 4 = 24x + 8 + 4 = 24x + 12.

Now let's evaluate g(f(x)) = g(8(x - 3) + 4).

Substitute 8(x - 3) + 4 into g(x) and simplify:

g(f(x)) = 3(8(x - 3) + 4) + 4 = 3(8x - 24 + 4) + 4 = 3(8x - 20) + 4 = 24x - 60 + 4 = 24x - 56.

Since f(g(x)) = x and g(f(x)) = x, we can conclude that the pair of functions (c) f(x) = 8(x - 3) + 4 and g(x) = 3x + 4 are inverses of each other.