Question

Consider the function and its inverse.

and

When comparing the functions using the equations, which conclusion can be made?

The domain of f(x) is restricted to x ≥ 0, and the domain of f–1(x) is restricted to x ≥ 0.
The domain of f(x) is restricted to x ≥ 0, and the domain of f–1(x) is restricted to x ≤ 0.
The domain of f(x) is restricted to x ≤ 0, and the domain of f–1(x) is restricted to x ≥ 4.
The domain of f(x) is restricted to x ≤ 0, and the domain of f–1(x) is restricted to x ≤ 4.

Answer 1

I don't see the two equations in your message — please paste the expressions for f(x) and f⁻¹(x) (or a picture). Without them I can only give the rule and an example.

Rule you should use:
- The domain of f⁻¹ is the range of f, and the domain of f is the range of f⁻¹.
- To find the range of f algebraically, solve y = f(x) for x and see what y-values are allowed (equivalently, find the domain of the inverse).
- If f involves a square root, log, etc., use the usual real-number restrictions (radicand ≥ 0, argument > 0 for logs, etc.).

Example (common case that matches one answer choice):
- If f(x) = √x then domain(f) is x ≥ 0. Solve y = √x → x = y², so f⁻¹(x) = x² but since y = √x ≥ 0 the inverse’s domain is x ≥ 0. So domain(f) ≥ 0 and domain(f⁻¹) ≥ 0 (first choice).

Send the two equations and I will pick which of the four choices is correct and show the steps.