# Video: Defining Inverse Functions by Restricting Domains

Liam is looking for an inverse to 𝑓(𝑥) = −2 − (𝑥 − 4)². He starts with the parabola 𝑦 = −2 − (𝑥 − 4)². He then reflects this in the line 𝑦 = 𝑥 to get the shown parabola 𝑥 = −2 − (𝑦 − 4)². Complete Liam’s work by determining the inverse 𝑓⁻¹ whose graph is the given solid curve.

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### Video Transcript

Liam is looking for an inverse to the function 𝑓 of 𝑥 equals negative two minus 𝑥 minus four squared. He starts with the parabola 𝑦 equals negative two minus 𝑥 minus four squared. He then reflects this in the line 𝑦 equals 𝑥 to get the shown parabola 𝑥 equals negative two minus 𝑦 minus four squared. Complete Liam’s work by determining the inverse whose graph is the given solid curve.

Liam already went ahead and took the first step of replacing 𝑓 of 𝑥 with 𝑦. He also went ahead and already interchanged 𝑥 and 𝑦. Now what we need to do is to solve for 𝑦. So first, let’s add two to both sides of this equation. Next, let’s divide by negative one. Our next step to solve for 𝑦 would be to square root both sides. So we have the square root of negative 𝑥 minus two is equal to 𝑦 minus four. And now we need to add four to both sides. Therefore, 𝑦 is equal to four plus the square root of negative 𝑥 minus two.

And now our last step is to replace 𝑦 with 𝑓 inverse. So after replacing 𝑦 with 𝑓 inverse, we put it on the left. And now our answer is: 𝑓 inverse is equal to four plus the square root of negative 𝑥 minus two.