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Finding an Inverse Graphically - Problem 2

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Finding the inverse of a function graphically. So whenever we find the inverse of a function the first thing we want to do is check if our function is one-to-one, so does it pass the horizontal line test? This function we draw a horizontal line anywhere it only hits one point so this function is one-to-one.

To find the inverse graphically, the first thing we always want to do is sketch the line y equals x. So line y equals x is just a slip of one to the origin, something roughly like that. So what I do from here is this line y equals x so on this line x and y are the same when we find the inverse we switch our x and our y values so any point that is on this line is going to remain on this line. Our x and y values are the same, we switch them we end up with the same point.

So I know that this point and these two points are going to be on the inverse as well because we flipped them they stay the same. What I did then is just draw some key points and knowing that the distance perpendicularly from the line y equals x is going to remain the same. So this point right here is about an inch below the line y equals x. When I flipped the y and x values what’s going to happen is that's going to translate to a point about the same distance on the other side of the curve.

Doing the same thing, what I sort of call key points anything where the graph sort of does something dramatic and then connecting the dots to create a new graph. We're doing the same thing over here. This point is sort of our key point. This distance about this we flip that over we end up with our new point roughly down here it’s not a complete signs but we got a rough idea of what’s happening.

So we know that this segment right here is going to look roughly, oops if I can draw a straight line it would help, roughly like that. So then we know that this is a straight segment so this line is going to connect through like that. We are doing the same thing for this little point right here rough to the same distance over, connect these points and then the same thing for our main and our last endpoint same distance perpendicular roughly here finish that line.

It’s not precise but it gives us a really good idea of what the inverse is going to look like. So whenever you have a graph draw your line y equals x and just flip points over that any key points where the graph changes directions it’s a really good spot, we can connect the dots to get the idea of what’s happening.

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