# Question Video: Defining Inverse Functions by Restricting Domains Mathematics • 10th Grade

Liam is looking for an inverse to π(π₯) = β2 β (π₯ β 4)Β². He starts with the parabola π¦ = β2 β (π₯ β 4)Β². He then reflects this in the line π¦ = π₯ to get the shown parabola π₯ = β2 β (π¦ β 4)Β². Complete Liamβs work by determining the inverse πβ»ΒΉ whose graph is the given solid curve.

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### Video Transcript

Liam is looking for an inverse to the function π of π₯ equals negative two minus π₯ minus four squared. He starts with the parabola π¦ equals negative two minus π₯ minus four squared. He then reflects this in the line π¦ equals π₯ to get the shown parabola π₯ equals negative two minus π¦ minus four squared. Complete Liamβs work by determining the inverse whose graph is the given solid curve.

Liam already went ahead and took the first step of replacing π of π₯ with π¦. He also went ahead and already interchanged π₯ and π¦. Now what we need to do is to solve for π¦. So first, letβs add two to both sides of this equation. Next, letβs divide by negative one. Our next step to solve for π¦ would be to square root both sides. So we have the square root of negative π₯ minus two is equal to π¦ minus four. And now we need to add four to both sides. Therefore, π¦ is equal to four plus the square root of negative π₯ minus two.

And now our last step is to replace π¦ with π inverse. So after replacing π¦ with π inverse, we put it on the left. And now our answer is: π inverse is equal to four plus the square root of negative π₯ minus two.