Purpose: Just as invNorm accepts a quantile value (between 0 and 1, inclusive) and returns a z score, the INVT program accepts a value between 0 and 1, plus user-specified degrees of freedom (df), and returns a t score. INVT thus fills a gap that TI left when designing the firmware of the TI-83.
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Graph an Inverse Function
Summary:After you graph a function on yourTI-83/84, you can make a picture of its inverse by using the
DrawInv command on the DRAW menu.
For this illustration, let’s usef(x) = √(x−2), shown at right.Though you can easily find the inverse of this particular functionalgebraically, the techniques on this page will work for anyfunction.
I’ve compensated for the rectangular viewing window bysetting window margins to 0 to 10 in the x direction and 0 to 6.5 inthe y direction.(If you don’t know how to graph a function, pleasereview that procedure.)
The graph of an inversefunction is a mirror image of the original through the line y=x, soI’ve also plotted that line.
To draw the inverse of that 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 thegraph of f-1(x), and f-1 Download visual foxpro 6.0. (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 seethat the inverse of the sample function appears only in the positive xregion. The inverse you calculate algebraically, x²+2, has a domainin both the positive and negative reals, but from drawing the inverseon the TI-83/84 you can see that you need to restrict the inversefunction’s domain to match the restricted range of the originalfunction.
There’s another way you can check your work. Find the inversefunction first, algebraically, and graph it as
Y3 whenyou graph the original as Y1 . If you do that,DrawInv Y1 will exactly overlay the graph of youralgebraic inverse.
Caution: Because the screen resolution is low, two differentfunctions sometimes look the same. This method isn’t an absoluteguarantee that your work is correct, but it’s better than nocheck at all.
Find the Value of an Inverse Function
Now suppose you have to find f-1(1.5)? Of course you canlook at it on the graph and estimate, but your calculator can do abetter job of the estimation for you. There are two methods, one onthe graph and one on the home screen.
Method 1:
|
Select the intersect command. |
[2nd TRACE makesCALC ] [5 ] |
The calculator asks “First curve?” |
Simply press [ENTER ] to select the firstcurve. |
The calculator then asks “Second curve?” | The cursor may have moved automatically to the other curve. Ifnot, press [▲ ] or [▼ ] until itdoes.Press [ ENTER ]. |
Finally, the calculator asks for your guess. | Usually you can just press [ENTER ]. But if the functionis very complicated, you can use [◄ ] or[► ] to move the cursor close to the intersectionpoint and then press [ENTER ], or type in a number andpress [ENTER ]. |
The result is shown at right: the answer is 4.25.
Leader Board Leading Today Pts Helpful 1. Update adobe flash player.
Why is the answer x and not y? Because you’re trying tofind 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 goodapproximation, 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 usingthe
solve
function.
Select the solve function from the catalogbecause it’s not in a menu. (There’s a Solver command in the Math menu, but setting it up is a little morework.) |
Press [2nd 0 makesCATALOG ] [ALPHA 4 makesT ], scroll up tosolve( , and press [ENTER ]. |
The first argument is an expression that you want to equateto zero. You actually want to equate Y1 to 1.5, which isthe same as equating Y1 −1.5 to 0. |
Press [VARS ] [► ] [1 ] [1 ] [− ] [1 ] [. ] [5 ] |
The second argument is the variable, x. | Press [, ] [x,T,θ,n ]. |
The last argument is your initial guess. Unless the functionis pretty complicated, it doesn’t matter what you enter here aslong as it’s in the domain of the function. For example, 0 wouldbe a bad choice for f(x) = √(x−2) becausef(0) is not a real number. Let’s use 6 as the initialguess. | Enter the initial guess and a close parenthesis[) ]. |
The screen is shown at right. The answer of 4.25 agrees with the graphical method.
Caution: Again, remember that this is anumerical solution and may not be exact.
What’s New
- 15 Feb 2009: Add section, suggested by Max Harwood, onfinding value of inverse function;edit the graphingsection extensively for clarity; drop “drawing” fromdocument title.
- (intervening changes suppressed)
- 7 June 2003: First publication, as “Drawing InverseFunctions on the TI-83”.
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