Consider the function f(x)=(x+3)^2−2. Restrict the domain of f(x) such that it has an inverse function, f^−1(x). Submit your answer in interval notation.

Question

For example, if the restricted domain is all values of x greater than or equal to 2, then your response for the domain should be [2,∞).

Answer 1

First we need to find what the domain of the inverse is, and to do that, we need to find the inverse function:

y = (x+3)² - 2

y + 2 = (x+3)²

√(y+2) = x+3

√(y+2) - 3 = x

So the inverse function is f^-1(x) = √(x+2) - 3. The domain of this function is restricted to where the contents of the radical is non-negative. So x + 2 ≥ 0, or x ≥ -2. Because of the definition of inverse functions, this also happens to be the range of our original function. So we can use this to find the domain.

(x+3)² - 2 ≥ -2

(x+3)² ≥ 0

x + 3 ≥ 0 or x + 3 ≤ 0

x ≥ -3 or x ≤ -3

So there are two ranges that we can restrict the domain to such that f(x) has an inverse: (-∞, -3] and [-3, ∞).