How do you graph inverse trig functions?

To graph any inverse function, you take the domain and range (the x and y coordinates) and flip them.

This means, for example, that if the point #(color(red)7, color(blue)2)# is on #f(x)#, then the point #(color(blue)2, color(red)7)# must be on #f^-1(x)#.

But, for this to work, the function must be one-to-one, meaning that there is only one x-value for each y-value in the range. To test if a function is one-to-one, we use the horizontal line test.

If you've ever seen the graph of a trig function, you know that the horizontal line test does NOT work, since if you try to place a horizontal line on the graph, it will cross the graph at infinitely many places.

For example, this horizontal line crosses #secx# 6 times just in this small slice of the graph:
graph{(y-secx)(y-4) = 0 [-10.045, 9.955, -0.48, 9.52]}

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So, how do we make the trig functions one-to-one so that we can graph their inverse functions?

Well, we actually need to cut out most of the trig function. In fact, the part of the function that we use to make the inverse will just be ONE wavelength, instead of infinitely many.

How do we know which part of the trig functions to use to make the inverse graph? The values were actually pre-defined by mathematicians before us. We must simply memorize the correct values to use. If you don't have these values memorized, here is a table showing which x-values to keep and flip when making the new graph:

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Now that we have these ranges, how do we make the inverse function? There are 3 steps:

1. Graph the original function
2. Isolate the part you need
3. Flip the x and y coordinates for that part

And that's all there is to it! If this explanation is still confusing to you, let me do an example to demonstrate the process. I will find the inverse function of #tan(x)#.

Step 1: Graph the original function

Here is the original graph of #tanx#:
graph{tanx [-9.96, 10.04, -4.6, 5.4]}

Step 2: Isolate the part you need

According to the table above, we only need to keep the part of the graph between #-pi/2# and #pi/2#.

So, let's get rid of everything else. This is the part that we will flip to make our inverse function:
graph{arctan(y) = x [-9.96, 10.04, -4.6, 5.4]}

Step 3: Flip the x and y coordinates

This is the same as flipping the function around the line #y=x#.

If you used coordinates to graph the original function, take these points and flip the two values. Draw these new points on another graph, and connect them to get the inverse function, which should look something like this:

graph{arctan(x) [-9.96, 10.04, -4.6, 5.4]}

This is the finished inverse graph. Hope this explanation helps!