What is the inverse of f if f(x)=3/sqrt x-5
A) f^-1 (x) = x^3 + 125
B) f^-1 (x) = x^3 + 5
C) f^-1 (x) = 1/ 3/sqrt x-5
D) f^-1 (x) = 5 - 3/sqrt x
The inverse of a function f(x) is denoted as f^(-1)(x) and is defined as the function that undoes the operation of f(x) on its input and gives back the original input.
Given f(x) = 3/sqrt(x - 5), let's find its inverse.
1. Start by expressing f(x) = y:
y = 3/sqrt(x - 5)
2. Swap x and y:
x = 3/sqrt(y - 5)
3. Solve for y:
x = 3/sqrt(y - 5)
x * sqrt(y - 5) = 3
sqrt(y - 5) = 3/x
y - 5 = (3/x)^2
y - 5 = 9/x^2
y = 9/x^2 + 5
Therefore, f^(-1)(x) = 9/x^2 + 5.
The correct answer is D) f^-1(x) = 5 - 3/sqrt(x).