Question

If f (x) = 5 x minus 25 and g (x) = one-fifth x + 5, which expression could be used to verify g(x) is the inverse of f(x)?

Answer 1

To verify that g(x) is the inverse of f(x), where f(x) = 5x - 25 and g(x) = ⅕x + 5, one needs to compute both f(g(x)) and g(f(x)) and show that each composition simplifies to the identity function, returning the value x.

Step-by-step explanation:

To verify that g(x) is the inverse of f(x), one must show that the composition of f and g, that is f(g(x)) and g(f(x)), both result in the original input x. Given f(x) = 5x - 25 and g(x) = ⅕x + 5, we calculate the composition of both functions.

First, substitute g(x) into f(x) to get f(g(x)) = f(⅕x + 5). Plugging in the expression for g(x) into f(x), we find:

f(g(x)) = 5(⅕x + 5) - 25

Next, simplify the expression:

f(g(x)) = x + 25 - 25

f(g(x)) = x

Similarly, substitute f(x) into g(x) to get g(f(x)) = g(5x - 25). Plugging in the expression for f(x) into g(x), we find:

g(f(x)) = ⅕(5x - 25) + 5

Continue simplifying:

g(f(x)) = x - 5 + 5

g(f(x)) = x

Since both f(g(x)) and g(f(x)) reduce to x, this confirms that g(x) is indeed the inverse of f(x).