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) = x2 + 1 and g(x) = x + 2.

Step-by-Step:

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) = x2 + 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) = x2 + 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) = 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.

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See Also

What is a function? Expanding brackets What is the FOIL method? Collecting like terms