Prior to starting this module you should review the Evaluating functions module: Evaluating Functions
Composition of functions is the same philosophy as evaluating functions for a given value of the variable. The difference is that Composition of Functions means that we will substitute one function into an equation instead of just a number. Then we will simplify the new function when possible. Another difference is you will be given 2 equations instead of one.
For example: given f(x) = 3x and g(x) = 4x, we will be asked to substitute one equation in to the other. The notation used for this is written one of two ways:
1. g(f(x))
2. f(g(x))
g(f(x)) tells us to use the function g(x) as our main function, then substitute for x the equation f(x).
f(g(x)) tells us to use the function f(x) as our main function, then substitute for x the equation g(x).
Example 1:
Given f(x) = 3x and g(x) = 4x, perform g(f(x)).
Solution: Use g(x) = 4x as your main equation, and substitute for x the equation f(x) = 3x.
g(f(x)) = 4(3x)
Now simplify the new equation. This will be our answer.
g(f(x)) = 12x
Using the same two equations, perform f(g(x)). Solution: Use f(x) as your main equation and substitute g(x) for x. Then simplify.
f(x) = 3x and g(x) = 4x, perform f(g(x)).
f(g(x)) = 3(4x)
Now simplify the equation. This will be the answer: f(g(x)) = 12x.
Example 2:
Given f(x) = 5 - x and g(x) = 6x + 2
Perform f(g(x)).
Solution: Use the f(x) equation and substitute the second equation in to the first for variable x.
f(g(x)) = 5 - (6x + 2)
Now simplify the equation: 5 - (6x + 2) ➾ 5 - 6x - 2 ➾ -6x + 5 - 2 ➾ -6x +3 *Answer
f(g(x)) = -6x + 3
Example 3:
Given f(x) = 5 - x and g(x) = 6x + 2, perform g(f(x)).
Solution: Use the g(x) equation and substitute the second equation in to the first for variable x.
g(f(x)) = 6(5 - x) + 2
Then simplify the new equation:
6(5 - x) + 2 ➾ 30 - 6x + 2 ➾ -6x + 30 + 2 ➾ -6x + 32
g(f(x)) = -6x + 32 *Answer
Example 4:
Given f(x) = x + 7 and g(x) = x2 - 3, perform f(g(x)).
Substitute the g(x) equation in to the f(x) equation.
f(g(x)) = (x2 -3) + 7
f(g(x)) = x2 - 3 + 7
f(g(x)) = x2 + 4 *Answer
Now you try an example. Enter your answer into the space provided, then check your answer.
Example 5: Given f(x) = x2 + 6 and g(x) = 5x, perform f(g(x)).
Use the caret key "^" (shift 6) to raise x to the second power (x2).Leave one space between each term.
For example: 5x2 + 9: type: 5x^2 + 9.
Answer
Example 6: Given f(x) = 3x + 6 and g(x) = x + 4, perform f(g(x)).
Leave one space between each term.
Answer
Example 7: Given f(x) = 2x and g(x) = x + 5, perform g(f(x)).
Substitute the f(x) equation in to the g(x) equation for x, then simplify. Leave one space between each term when typing in your answer.
Answer
Example 8: Given f(x) = 5x - 4 and g(x) = 2x + 5, perform g(f(x)).
Substitute the f(x) equation in to the g(x) equation for x, then simplify. Leave one space between each term when typing in your answer.
Answer