Find the inverse function

Question

f(x)=(x-3)²

For x ≤ 3, f(x) is one-to-one such that f ⁻¹(y) = ______ with y ≥ 0

Answer 1

Hello Anika, I sincerely apologize, I had it wrong previously. Please find the edited version here.

1st: the domain of f(x) is (-∞, 3]; the range of f(x) is [0, ∞). Keep in mind:

The domain of the inverse = range of original function, so domain of f^-1(y) = [0, ∞).

Range of inverse = domain of original function = (-∞, 3]

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Now, to find that inverse, you would replace f(x) by y and get: y = (x-3)²

Next, you'd need to solve for x. You could square root both sides of the previous equation and get:

±√y = x - 3

Add 3 to both sides and you get f^-1 (y) = x = 3 ±√y

However, your answer has to be one of 3 +√y or 3 -√y. Which of those two potential inverse functions have a domain of [0, ∞) and a range of (-∞, 3]

Both have [0, ∞) as domain but only 3 -√y has a range of (-∞,3], so the final answer is f ⁻¹(y) = 3 - √y