Inverse Functions

This section is currently under construction.

1. How can I plot the graphs of y=f(x), y=f ^-1(x), and y=x on the same set of axes?

Answer #1: We will use the parametric mode feature of the TI83 calculator. Press [MODE] and select "Par" on line 4. Then press [Y=] and set X_1T = T, Y_1T = f(T), X_2T = f(T), Y_2T = T, X_3T = T, Y_3T = T. These parametric equations will plot the graphs of y=f(x), x=f(y) (which is the graph of y=f ^-1(x) ), and y=x.

The difficult part is to get the best window settings. Try the following: Press [WINDOW]. Choose values for Xmin, Xmax, Ymin, Ymax, which satisfy the following conditions:

Condition#1. Ymin = Xmin.
Condition#2. Ymax = Xmin + (62/94)(Xmax - Xmin) .
Condition#3. A portion of the graph of Y = f(X) will appear on the graph screen for X in the interval [Xmin, Xmin + (62/94)(Xmax - Xmin)]. (This interval is the same as the interval [Ymin,Ymax].)

Conditions#1 and 2 will guarantee that the "graph grid" will be "square", and that part of the graph of Y = X will appear. Condition#3 will guarantee that part of the graph of Y = f ^-1(X) will appear.

Now set Tmin = Xmin, Tmax = Xmax, and Tstep = (Tmax - Tmin) / 70 . (These "T" settings will plot 71 points in the graph.) Then press [GRAPH]. You should be able to see the symmetry of the graphs with respect to the line Y = X.

(Roughly, the above procedure is based on the idea of setting up a window which contains a square, where the square has the line Y = X as a diagonal (conditions#1,2). Thus, if the graph of Y = f(X) appears anywhere in this square (condition#3), then the graph of Y = f ^-1(X) will also appear in the same square.)

Answer #2: Press [MODE] and select "Func" on line 4. Put f(X) into Y₁ and put X into Y₂. (Thus, Y₂ will contain the function y = x . )

Set up the graph window as in Answer #1. Again, this is the difficult part. (As a first attempt at window settings, you might want to try [ZOOM] [4] , the "ZDecimal" settings.)

Now press [GRAPH] to get the graphs of y = f(x) and y = x . Next, press [2nd] [DRAW] [8] [VARS] [Right Arrow] [ENTER] [ENTER] . This series of buttons will place the instruction "DrawInv Y₁" on the home screen.

Now press [ENTER]. The calculator will "Draw" the graph of y = f ^-1(x) onto the graph screen, together with the graphs of y = f(x) and y = x .






Related questions




Q1. How do I graph a function of y?

Q2. How can I compute a) cot ^-1(x) , b) sec ^-1(x) , and c) csc ^-1(x) using the TI-83 calculator?




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