Question

If f(x)=√x​−6, find the inverse of f(x) and its domain.

a) f−1(x)=(x+6)²,x≥6
b) f−1(x)=(x+6)²,x≤6
c) f−1(x)=(x−6)²,x≥6
d) f−1(x)=(x−6)²,x≤6

Answer 1

The inverse function of f(x) = √x - 6 is f^{-1}(x) = (x + 6)^2, with a domain of x ≥ -6.

Step-by-step explanation:

To find the inverse f-1(x) of the function f(x) = √x - 6, we must solve for the variable x such that y = √x - 6. To do this, we need to reverse the operations performed on x in f(x). We follow these steps:

1. Replace f(x) with y: y = √x - 6.
2. Swap x and y: x = √y - 6.
3. Solve for y by first adding 6 to both sides: x + 6 = √y.
4. Square both sides to get rid of the square root: (x + 6)2 = y.
5. The inverse function is y = (x + 6)2.

Since the original function has a square root, it only makes sense for the input to be non-negative (as we cannot take the square root of a negative number in the real number system). Therefore, the domain of the original function is x ≥ 0. After reflecting across the line y = x to find the inverse, the domain of the inverse function will be y ≥ 0.

Considering the range of the original function starts at -6 and goes to infinity, because square root functions always produce non-negative outputs, the domain of the inverse function is x ≥ -6. Hence, the correct answer is: