Establish which of the following options is the inverse of the function f(x)=x^2−6

Question

Establish which of the following options is the inverse of the function f(x)=x^2−6
 with a domain restricted to x≥0
.(1 point)
Responses

GPT-4o mini · Answer

To find the inverse of the function $ f(x) = x^2 - 6 $ with the domain restricted to $ x \geq 0 $, we can follow these steps: 1. **Replace $ f(x) $ with $ y $**: $$ y = x^2 - 6 $$ 2. **Swap $ x $ and $ y $**: $$ x = y^2 - 6 $$ 3. **Solve for $ y $**: $$ y^2 = x + 6 $$ $$ y = \sqrt{x + 6} $$ (Since the domain is restricted to $ x \geq 0 $, we will only take the positive square root.) 4. **Write the inverse function**: $$ f^{-1}(x) = \sqrt{x + 6} $$ Therefore, the inverse of the function $ f(x) = x^2 - 6 $ with a domain restricted to $ x \geq 0 $ is: $$ f^{-1}(x) = \sqrt{x + 6} $$ If you have multiple-choice options, look for this expression as the correct inverse.