Inverse Functions on TI-83/84

Graph an Inverse Function

Summary: After you graph a function on your TI-83/84, you can make a picture of its inverse by using the DrawInv command on the DRAW menu.

For this illustration, let’s use f(x) = √x−2, shown at right. Though you can easily find the inverse of this particular function algebraically, the techniques on this page will work for any function.

I’ve compensated for the rectangular viewing window by setting window margins to 0 to 10 in the x direction and 0 to 6.5 in the y direction. (If you don’t know how to graph a function, please review that procedure.)

The graph of an inverse function is a mirror image of the original through the line y = x, and here’s how to plot that inverse function:

The result is shown at right.

You know from your algebra work that the inverse of

f(x) = √x−2

is

f ⁻¹(x) = x²+2, x ≥ 0

and the graph confirms that.

Each point on the graph of f(x) has a corresponding point on the graph of f ⁻¹(x). For example, f(2) = 0, so (2,0) is on the original graph. (0,2) is on the graph of f ⁻¹(x), and f ⁻¹(0) = 2.

Unfortunately, all you can do with the inverse is look at it. You can’t trace or do other things. But even that helps you check your work. For instance, you see that the inverse of the sample function appears only in the positive x region. The inverse you calculate algebraically, x²+2, has a domain in both the positive and negative reals, but from drawing the inverse on the TI-83/84 you can see that you need to restrict the inverse function’s domain to match the restricted range of the original function.

There’s another way you can check your work. Find the inverse function first, algebraically, and graph it as Y3 when you graph the original as Y1. If you do that, DrawInv Y1 will exactly overlay the graph of your algebraic inverse.

Caution: Because the screen resolution is low, two different functions sometimes look the same. This method isn’t an absolute guarantee that your work is correct, but it’s better than no check at all.

Find the Value of an Inverse Function

Now suppose you have to find f ⁻¹(1.5). (This is equivalent to asking how you can find the x that corresponds to a given y.) Of course you can look at it on the graph and estimate, but your calculator can do a better job of the estimation for you. There are two methods, one on the graph and one on the home screen.

Method 1: intersect on Graph

Remember that f ⁻¹(1.5) is some value, call it a, such that (1.5,a) is on the graph of f ⁻¹(x), and therefore (a,1.5) is on the graph of f(x). In other words, f ⁻¹(1.5) is the x value on the original graph of f(x) where the y value is 1.5.

Using this idea, to find f ⁻¹(1.5) you can plot y = 1.5 and have your calculator find the point where it intersects the graph of f(x). You don’t need the graph of f ⁻¹(x) for this at all.

The graphs are shown at right, and here’s the procedure.

Select the intersect command.	[2nd TRACE makes CALC] [5]
The calculator asks “First curve?”	Simply press [ENTER] to select the first curve.
The calculator then asks “Second curve?”	The cursor may have moved automatically to the other curve. If not, press [▲] or [▼] until it does.   Press [ENTER].
Finally, the calculator asks for your guess.	Usually you can just press [ENTER]. But if the function is very complicated, you can use [◄] or [►] to move the cursor close to the intersection point and then press [ENTER], or type in a number and press [ENTER].

The result is shown at right: the answer is 4.25.

Why is the answer x and not y? Because you’re trying to find f ⁻¹(1.5), the value of the inverse function of 1.5. But as mentioned above, f ⁻¹(1.5) is the number a such that f(a) = 1.5. In other words, because f(4.25) = 1.5, f ⁻¹(1.5) = 4.25.

Caution: Your calculator gives numerical solutions only. To determine whether 4.25 is the exact answer or just a good approximation, you have to check it in the original function.

Method 2: solve on Home Screen

You can accomplish the same thing on the home screen by using the solve function.

The screen is shown at right. The answer of 4.25 agrees with the graphical method.

Caution: Again, remember that this is a numerical solution and may not be exact.