The LessonA composite function is a function of a function. A composite function combines two or more functions so that the output of one function becomes the input of another.
How to Find a Composite FunctionFinding a composite function is easy.
QuestionFind the composite function fg(x), where f(x) = x2 + 1 and g(x) = x + 2.
Understand the composite function. In our example, the composite function is fg(x). Reading right to left, this means:
- the input x is passed into the function g
- which is passed into the function f
Find the left most letter. In our example, the composite function is fg(x). f is the left most letter. Write out the function f(x)
f(x) = x2 + 1
We are passing g(x) into this function
- Insert a g to the right of the f in the function name.
fg(x) = x2 + 1
- Replace x with g(x).
fg(x) = g(x)2 + 1
Substitute g(x) = x + 2 into the function.
fg(x) = (x + 2)2 + 1Don't forget: Use brackets when substituting an expression. This reminds you that the whole expression is being squared.
Simplify the equation.
|fg(x) = (x + 2)2 + 1|
|fg(x) = (x + 2)(x + 2) + 1||(x + 2)2 = (x + 2) × (x + 2)|
|fg(x) = x2 + 4x + 4 + 1||Use the FOIL method to expand the brackets|
|fg(x) = x2 + 4x + 4 + 1||Collect the constant terms|
|fg(x) = x2 + 4x + 5|
Answer:The composite function fg(x) of f(x) = x2 + 1 and g(x) = x + 2 is: fg(x) = x2 + 4x + 5.
Lesson SlidesThe slider below shows another real example of how to find a composite function. Open the slider in a new tab
A Note on NotationThe image below shows a composite function gf(x), where a function f is passed to a function g: The composite function is gf(x).
- f is applied to the input x.
- g is applied to the function f.