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Problem 2 Medium Difficulty

(a) Suppose $ f $ is a one-to-one function with domain $ A $ and range $ B $. How is the inverse function $ f^{-1} $ defined? What is the domain of $ f^{-1} $? What is the range of $ f^{-1} $?
(b) If you are given a formula for $ f $, how do you find a formula for $ f^{-1} $?
(c) If you are given the graph of $ f $, how do you find the graph of $ f^{-1} $?

Answer

a) $f^{-1}(a)=b \operatorname{IF} f(b)=a$
Range of $f=$ Domain of $f^{-1}$
Domain of $f=$ Range of $f^{-1}$
b)
1) Write $y=f(x)$ .
2) Solve this equation for $x,$ in terms of $y$ (if possible).
3) Interchange $x$ and $y$ to express $f^{-1}$ as a function of $x .$ Thus, $y=f^{-1}(x)$
c)
The graph of $f^{-1}$ is obtained by reflecting the graph of $f$ about the line
$y=x$ .

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Video Transcript

All right, let's define the inverse of F of X. So for every point with coordinates, a B on F the point be a is on F in verse now because we switch the coordinates the X and Y coordinates, we end up also switching the domain and range. So for F in verse, the domain will be be and the range will be a. Now let's look at how you find the formula for an inverse. So if you're given the formula for F, what you want to do is first switch or interchange X and Y in your formula interchange. The places put X where, why was put why, where x waas and then solve for Y. This will give you the inverse, and then you can also rename it as F inverse when you're done. Okay, Now, what about graphically? If you're given the graph of F, how do you find the graph of F in verse? It's going to be the reflection of F a cross. The line y equals X. So, for example, let's say your graph f waas something like this. And let's say the line y equals X was here then, when you reflect f a cross the line y equals X. You're going to get something like that and that would be f in verse.