Problem:
Given the functions

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

Visualization:
Using Microcalc:
  1. Choose Beginning Calculus from the initial menu and press .
  2. Use the arrow keys to move to the menu item Graph y = F(x), press and .
  3. At the prompt, F(x) =, enter the function
    1/(1 + x^2)
  4. and use the bounds
    x0 = -3 x1 = 3
    y0 = -1.2 y1 = 1.2
  5. You will then see the graph of this function.
  6. Press and, using the arrow keys, move to the menu item Superimpose and press .
  7. At the prompt F(x) = , enter
    sin(x)
  8. You will then see the graphs of both functions on the screen.
  9. Press and, using the arrow keys, move to the menu item, Function Editor, and press .
  10. We will now define the composition fg. You will notice that the formula for g has been copied to F1. For reasons that will become apparent soon, let us reenter the formula for f; using the arrow keys, select Define F3 and press .
  11. Type
    1/(1 + x^2)
  12. Now, using the arrow keys, select Compose and press .
  13. Choose Outer Function, press , and then, using the arrow keys, pick F3 and press .
  14. Choose Inner Function, press , and, using the arrow keys, pick F1 .
  15. Choose Execute .
  16. You will then see the composition:
    F3 = 1/((Sin x) ^2 + 1).
  17. To graph this, press and, using the arrow keys, move to the menu item Superimpose and press .
  18. At the prompt F(x) =, press . You will then see the graphs of the composition and the two original functions.
  19. Finally, to get the graph of gf, we need to repeat this procedure. Press and, using the arrow keys, move to the menu item, Function Editor, and press .
  20. Note that the formula for fg has been copied to F1. Now, using the arrow keys, select Compose .
  21. Choose Outer Function and then, using the arrow keys, pick F2 and press .
  22. Choose Inner Function, , and then, using the arrow keys, pick F3 .
  23. Choose Execute . You will then see the composition:
    F2 = Sin[1 /(x ^2 + 1)]
  24. Press and, using the arrow keys, move to the menu item Superimpose .
  25. At the prompt F(x) =, press . You now have the graphs of all four functions as shown in the figure below.