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

Liam is looking for an inverse to ๐‘“(๐‘ฅ) = โˆ’2 โˆ’ (๐‘ฅ โˆ’ 4)ยฒ. He starts with the parabola ๐‘ฆ = โˆ’2 โˆ’ (๐‘ฅ โˆ’ 4)ยฒ. He then reflects this in the line ๐‘ฆ = ๐‘ฅ to get the shown parabola ๐‘ฅ = โˆ’2 โˆ’ (๐‘ฆ โˆ’ 4)ยฒ. Complete Liamโ€™s work by determining the inverse ๐‘“โปยน whose graph is the given solid curve.

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Video Transcript

Liam is looking for an inverse to the function ๐‘“ of ๐‘ฅ equals negative two minus ๐‘ฅ minus four squared. He starts with the parabola ๐‘ฆ equals negative two minus ๐‘ฅ minus four squared. He then reflects this in the line ๐‘ฆ equals ๐‘ฅ to get the shown parabola ๐‘ฅ equals negative two minus ๐‘ฆ minus four squared. Complete Liamโ€™s work by determining the inverse whose graph is the given solid curve.

Liam already went ahead and took the first step of replacing ๐‘“ of ๐‘ฅ with ๐‘ฆ. He also went ahead and already interchanged ๐‘ฅ and ๐‘ฆ. Now what we need to do is to solve for ๐‘ฆ. So first, letโ€™s add two to both sides of this equation. Next, letโ€™s divide by negative one. Our next step to solve for ๐‘ฆ would be to square root both sides. So we have the square root of negative ๐‘ฅ minus two is equal to ๐‘ฆ minus four. And now we need to add four to both sides. Therefore, ๐‘ฆ is equal to four plus the square root of negative ๐‘ฅ minus two.

And now our last step is to replace ๐‘ฆ with ๐‘“ inverse. So after replacing ๐‘ฆ with ๐‘“ inverse, we put it on the left. And now our answer is: ๐‘“ inverse is equal to four plus the square root of negative ๐‘ฅ minus two.

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