Find the inverse of the function on the given domain.

Question

Find the inverse of the function on the given domain. 

f(x)=(x-2)^2,[2,infinity)

Answer 1

f(x) = y = (x - 2)²

Switch x and y to get x = (y - 2)²

Solve for y: y - 2 = √x (technically there is a ± in front of the radical, but since x≥2 and we switched x and y, we have y≥2. So, y - 2≥0. Therefore, a minus sign in front of the radical makes no sense.

So, y = f^-1(x) = √x + 2

Answer 2

Hello Amber,

So we have,

f(x) = (x - 2)²

with a domain = [2,∞)

The inverse function is one the will take the output of f(x) as an input and give you back x as its output.

If f(x) = (x - 2)²,

then the inverse of f(x) is,

f(f(x)) = x

To get the latter expression we just have to use some algebra to get x alone.

(x - 2)² = f(x)

x - 2 = √f(x)

x = √f(x) + 2

f(f(x)) = √f(x) + 2

so the inverse function of y(x) would be,

f(f(x)) = √f(x) + 2

Let's say x was equal to 3, then,

f(3) = (3 - 2)²

f(3) = 1

and the inverse function would be,

f(f(3)) = √f(3) + 2

f(f(3)) = √1 + 2

f(f(3)) = 1 + 2

f(f(3)) = 3

You can see that in the first function we substituted 3 for x and got 1 as the output and in the second function we substituted 1 for f(x) and got back 3.

The domain is all of the x (input) values that are defined for the function. The inverse function has x as its output, so the domain of the original function becomes the range, or all the possible output values, of the inverse function.

The domain of f(x) = [2,∞)

The range of f(f(x)) = [2,∞)

The domain of f(f(x)) is the range of f(x) so first, we need to figure out the minimum and maximum y values possible for f(x)

The smallest x can be is 2,

f(2) = (2 - 2)²

f(2) = 0

and the largest number is infinity,

f(∞) = (∞ - 2)²

f(∞) = ∞

Since f(x) has a range of [0,∞), f(f(x)) has the domain [0,∞)

Therefore the inverse function of f(x) = (x - 2)² on the domain = [2,∞) is,

f(f(x)) = √f(x) + 2

on the domain [0,∞)

Let me know if this helped or if you need any further clarification!