Question

What is the inverse function of f(x) = (x - 5)² for x ≥ 5? Let function g represent the inverse of function f. Which of the following options correctly expresses g(x)?

1. g(x) = √x + 5, for x ≥ -5
2. g(x) = √x - 5, for x ≥ 5
3. g(x) = √(x - 5), for x ≥ 0
4. g(x) = x√5, for x ≥ 0

Please choose the option that accurately represents the inverse function g(x) corresponding to f(x) = (x - 5)² for x ≥ 5.

Answer 1

The inverse function of f(x) = (x - 5)² for x ≥ 5 is g(x) = √(x - 5), for x ≥ 0.

Step-by-step explanation:

The inverse function of f(x) = (x - 5)² for x ≥ 5 is the function g(x) that, when composed with f(x), yields the identity function. In other words, for any x ≥ 5, if f(x) = y, then g(y) = x. To find the inverse, we replace f(x) with y to get y = (x - 5)² and then solve for x by taking the square root of both sides. However, since x ≥ 5, we select the positive branch of the square root, yielding x = √(y - 5). Thus, the inverse function is g(x) = √(x - 5) with the domain x ≥ 0, making the correct answer option 3. So, the correctly expressed inverse function is g(x) = √(x - 5), for x ≥ 0.