Find the inverse of f informally . Verify that f(f ^ (- 1) * (x)) = x and f ^ (- 1) * (f(x)) = x f(x) = sqrt(x)

Question

You are verifying f(x)= square root x

Answer 1

We are starting out with f(x) = √x. Note that this function has a domain of x ≥ 0 and range of y ≥ 0.

To find the inverse function, switch the x and y (f(x)) and re-solve for y:

x = √y square both sides

y = x²

f^-1(x) = x², however we need to consider the domain and range

Since the range of f(x) was restricted to y ≥ 0 that means that the domain of f^-1(x) is restricted to x ≥ 0 as well.

So technically f^-1(x) = x², for x ≥ 0 (just the right half of the parabola)

Next, you are asked to use composition of functions to verify.

f(f^-1(x)) = √(x²) = x when x ≥ 0

f^-1(f(x)) = (√x)² = x when x ≥ 0

Therefore, they are inverses on the restricted domain.