Given the functions

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

Using Microcalc:

- Choose
*Beginning Calculus*from the initial menu and press . - Use the arrow keys to move to the menu item
*Graph y = F(x)*, press and . - At the prompt,
*F(x) =*, enter the function1/(1 + x^2) - and use the bounds
- x0 = -3 x1 = 3
- y0 = -1.2 y1 = 1.2

- You will then see the graph of this function.
- Press and, using the arrow
keys, move to the menu item
*Superimpose*and press . - At the prompt
*F(x) =*, entersin(x) - You will then see the graphs of both functions on the screen.
- Press and, using the arrow
keys, move to the menu item,
*Function Editor*, and press . - 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 . - Type
1/(1 + x^2) - Now, using the arrow keys, select
*Compose*and press . - Choose
*Outer Function*, press , and then, using the arrow keys, pick*F3*and press . - Choose
*Inner Function*, press , and, using the arrow keys, pick*F1*. - Choose
*Execute*. - You will then see the composition:
F3 = 1/((Sin x) ^2 + 1). - To graph this, press and, using the
arrow keys, move to the menu item
*Superimpose*and press . - At the prompt
*F(x) =*, press . You will then see the graphs of the composition and the two original functions. - 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 . - Note that the formula for
*fg*has been copied to F1. Now, using the arrow keys, select*Compose*. - Choose
*Outer Function*and then, using the arrow keys, pick*F2*and press . - Choose
*Inner Function*, , and then, using the arrow keys, pick*F3*. - Choose
*Execute*. You will then see the composition:F2 = Sin[1 /(x ^2 + 1)] - Press and, using the arrow keys,
move to the menu item
*Superimpose*. - At the prompt
*F(x) =*, press . You now have the graphs of all four functions as shown in the figure below.