Algebra Tutorials!
Friday 12th of July
Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# Composition of Functions

Example 1

The graphs of f(x) and g(x) are shown. Use these graphs to find (g f)(3).

Solution

Since (g f)(3) = g[f(3)], we first must find f(3). On the graph, locate f(3). That is, find the y-value of f when x = 3.

From the graph, we see that when x = 3, y = 6. Thus, f(3) = 6.

Now, use 6 as the input for g(x). That is, find g(6). To do this, find the y-value of g when x = 6.

From the graph we see that when x = 6, y = 2. Therefore, (g f)(3) = 2.

Example 2

Graph the function f(x) = x2 - 2. If g(x) = |x|, sketch the graph of (g f)(x).

Solution

The graph of f(x) = x2 - 2 is the same as the graph of f(x) = x2 but shifted down 2 units.

 Now, find (g ○ f)(x). Replace f(x) with x2 - 2. In g(x), replace x with x2 - 2. (g ○ f)(x) = g[f(x)]= g[x2 - 2] = | x2 - 2 |

The graph of g(x) = | x2 - 2 | is the same as f(x) = x2 - 2 except that all outputs are nonnegative because of the absolute value symbols.

To graph (g f)(x) = | x2 - 2 |, we can reflect across the x-axis the part of the graph of f(x) = x2 - 2 that is below the x-axis.

The graph of (g f)(x) = | x2 - 2 | lies on and above the x-axis.