Problem:
Given the functions

draw the graphs of the functions f, g, fg, and gf.

Solution:
Using TI-85 graphing calculator:

[Press here to see animation again!]

1. If you are not familiar with the graphing of functions on the TI-85, then first read the Initial Setup page from Little's Basic Guide to the TI-85.
2. Press the GRAPH key and then pick y(x)= by pressing the F1 key.
3. If necessary, keep pressing F4 key until only y1= appears on the screen.
4. At the prompt y1=, type in

1/(1+x^2)

and press the ENTER key.

5. At the prompt y2=, type in

sin x

and press the ENTER key.

6. At the prompt y3=, type in

1/(1+y2^2)

and press the ENTER key.

7. At the prompt y4=, type in

sin y1

and press the ENTER key.
IMPORTANT: Both of these y's must be lower case.

8. The range settings for the following graph is  [-3, 3] × [-2, 2].

9. Note that the graphs of the functions are plotted in the order in which they are entered above. You can plot only one or two of the graphs by using the SELCT key on the y(x)= screen. Since there seems to be only three graphs, the graphs of y1 and y4 were selected and the plot is the following:

You can see that there appears to be considerable overlap of these two graphs.

Alternative:

1. At the prompt y3=, type in

evalF(y1, x, y2)

and press the ENTER key.

2. At the prompt y4=, type in

evalF(y2, x, y1)

and press the ENTER key.

You can get the evalF command by pressing 2nd CALC and then the F1 key.