The inverse of a function is the same as the original function, but with the independent and dependent variables switched, which just means switch the x and y values in the coordinates.
The inverse of a function and the original will look the same, except that again, those x and y values will be switched, which means that they will be symmetric, the two graphs will be symmetric about the line y = x.
Next Lesson: Graphs of Functions
The transcript is provided for your convenience
Let’s look at an example. Graph y = 2x+3 and the inverse.
y = 2x+3 is a linear equation, and it’s in slope intercept form, y = mx+b. The m is our slope, so our slope is 2, or 2/1, and our y-intercept is 3, b is 3.
To graph a linear equation, we start by graphing the y-intercept. So, on our y-axis, we go up to 3, and there’s the first point for our line.
The slope is our rise and our run, so we need to rise 2 and run 1 from that point. Rise 2, run 1.
Now, we just need to connect these points with a line. And then put arrows on the end of our line to signify that line goes on forever.
So, that’s the graph of our original function, but how about the inverse? Remember from just a minute ago that the inverse of a function is just where the independent and dependent variables are switched, the independent variable being the x, and the dependent being the y. So, one way we could graph our inverse function is just to take our x and y coordinates from each one of these points and switch them.
So, this ordered pair is (0,3), which means that our inverse function will have the ordered pair (3,0). Just switch the x and y.
So, over 3, and then up 0. There’s the first point for our inverse function.
Then we can take this other point we put on our graph, which is (1,5) and do the same thing. Just switch the x and y coordinates, which makes it (5,1). So, over 5, and up 1.
We could’ve done this with any two points from our original function, but since these are the two points we used to graph our line, they were easy to use to switch the x and y coordinates. Now that we have our two points for our inverse function, we can connect those with a line, and put arrows on the end to signify that it goes on and on and on forever.
And that’s one way to graph an inverse function, to graph the original and then take the ordered pairs from the original and switch them.
The other way to graph an inverse function, function is to take the original function, and again, switch the x and y. So, the y becomes x, and the x becomes y.
Now, if we wanted to graph this, we’d want to put it in a form like slope intercept form, which means we must solve for y now. So, start by subtracting 3 from both sides, x-3 is equal to 2y, plus 3 minus 3 are added to the inverses so they cancel, and then we divide both sides by 2 to solve for y. 2 divided 2 is 1, and 1 times y is y, I’m going to use the symmetric property to flip this around so we get y = x-3/2.
Or, y = x/2 – 3/2, or y = 1/2x – 3/2, you can write it in lots of different ways, but then you could graph this inverse function the same way we graphed our original function by graphing the y-intercept, -1/2, and then we would use our slope, 1/2, 2 rise 1, and run 2. So, when you’re graphing a function and its inverse, just remember, switch the x and y.
Next Lesson: Graphs of Functions