Let f(x) = (x-7)^2 Make the function one-to-one by restricting the domain to those values greater than or equal to 7. State this domain.

Question

Let f(x) = (x-7)^2

Make the function one-to-one by restricting the domain to those values greater than or equal to 7. State this domain. ___________

Find the inverse of f  restricted to this domain

f^-1(x)= ___________

Answer 1

The domain is [7, ∞), or { x ∈ R : x ≥ 7 }.

The inverse of f restricted to this domain is f^-1(x) = x^1/2 + 7.

To see this, let g(x) = x². When x ≥ 0, g has inverse h(x) = x^1/2.

Since f(x) = g(x-7), then f^-1(x) = h(x) + 7 = x^1/2 + 7.

We can verify that, if x ≥ 7, then

f^-1(f(x)) = f^-1((x - 7)²) = ((x-7)²)^1/2 + 7 = x - 7 + 7 = x,

and if x ≥ 0, then

f(f^-1(x)) = f(x^1/2 + 7) = (x^1/2 - 7 + 7)² = (x^1/2)² = x