How to Find a Composite Function
Finding a Composite Function
A 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 Function
Finding a composite function is easy.
Question
Find the composite function fg(x), where f(x) = x^{2} + 1 and g(x) = x + 2.
StepbyStep:
1
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
2
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) = x^{2} + 1
3
We are passing g(x) into this function

Insert a g to the right of the f in the function name.
fg(x) = x^{2} + 1

Replace x with g(x).
fg(x) = g(x)^{2} + 1
4
Substitute g(x) = x + 2 into the function.
fg(x) = (x + 2)^{2} + 1
Don't forget: Use brackets when substituting an expression. This reminds you that the whole expression is being squared.
5
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) = x^{2} + 4x + 4 + 1  Use the FOIL method to expand the brackets 
fg(x) = x^{2} + 4x + 4 + 1  Collect the constant terms 
fg(x) = x^{2} + 4x + 5 
Answer:
The composite function fg(x) of f(x) = x^{2} + 1 and g(x) = x + 2 is:
fg(x) = x^{2} + 4x + 5.