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Finding an Inverse Graphically - Concept

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!

In order to understand graphing inverse functions, students should review the definition of inverse functions, how to find the inverse algebraically and how to prove inverse functions. The graphs of inverse functions and invertible functions have unique characteristics that involve domain and range. Techniques for graphing inverse functions can make it easier to graph certain functions by hand.

Finding the inverse of a funtion graphically. So for this example we are going to look at the graphic interpretation of what an inverse means. Okay, so behind me I have a function that is defined solely as 2 points to keep our life a little bit simple. Okay?
So what I want to do is plot these points. So we have the point 2, 4. 2, 4. And we also have the point -1, 3. Okay. So rough points right there. Now what I want to do with these is find the inverse. Okay? so remember whenever we find the inverse we just switch our x and our y value. So our inverse then is going to contain the points 4, 2 and the point 3, -1. Okay? Plotting those as well. We now go over 4 up 2 and over 3 down 1. And basically what happens whenever we plot the inverses is we reflect everything over this line y=x, okay? So what that looks like is if you'd sketch in the line y=x, [IB] just the slip of 1 through the origin. Every point is gets reflected or [IB] over that. So this point here just gets flipped over to that. this point here flipped over to that. The easiest way for me to sort of draw these out is to draw perpendicular lines. If you just wanted to find this without finding the actual point, draw perpendicular lines and the distance l is going to be roughly the same distance to that point. Okay? So just draw a perpendicular segment, flip it over and that point should end up reflected over the line y=x.
So finding a graph of an inverse function just by flipping over the line y=x.

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