Precalculus Combining and Composing Functions Lesson (Free PDF)

Precalculus math lesson: Combining and Composing Functions – Maths PDF

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This function operations lesson for Precalculus covers combining and composing functions (Common Core F-BF.A.1b, F-BF.A.1c), with a free PDF to print.

Composition applies one function to the output of another: (f∘g)(x) = f(g(x)) means evaluate g first, then f. Functions can also be added, subtracted, multiplied or divided at each x. The order of composition usually matters, and the domain of f(g(x)) needs x in the domain of g and g(x) in the domain of f.

Precalculus · Common Core State Standards: F-BF.A.1b, F-BF.A.1c

Key facts in this function operations lesson

Key facts

(f+g)(x) = f(x) + g(x); the same pattern holds for difference and product.
(f/g)(x) = f(x)/g(x) requires g(x) not equal to 0.
(f∘g)(x) = f(g(x)): work from the inside out.
In general f(g(x)) is not equal to g(f(x)).
Domain of f(g(x)): x must be in the domain of g, and g(x) in the domain of f.

How do you combine two functions?

Functions can be added, subtracted, multiplied and divided by combining their outputs at the same input. \((f+g)(x)=f(x)+g(x)\), \((f-g)(x)=f(x)-g(x)\), \((fg)(x)=f(x)\cdot g(x)\), and \(\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}\) with \(g(x)\neq 0\). The domain of the result keeps only the \(x\)-values that are in both domains, and a quotient also removes every \(x\) where \(g(x)=0\).

Composition

The composition \((f\circ g)(x)=f(g(x))\) means: apply \(g\) first, then apply \(f\) to the result. Read it ‘f of g of x.’

In \(f(g(x))\), the inner function \(g\) acts first and the outer function \(f\) acts second.

How do you evaluate a composite function?

Work from the inside out. If \(f(x)=3x-2\) and \(g(x)=x^{2}+1\), then \(g(2)=5\) and \(f(g(2))=f(5)=13\). To get a formula, substitute the whole expression \(g(x)\) wherever \(f\) has an \(x\): \(f(g(x))=3(x^{2}+1)-2=3x^{2}+1\).

A table works the same way. In the table below, \(g(1)=0\) and \(f(0)=2\), so \(f(g(1))=2\).

Values of f and g for x = 0, 1, 2, 3.

Does the order of composition matter?

Usually yes. With \(f(x)=x^{2}\) and \(g(x)=x+2\), we get \(f(g(x))=(x+2)^{2}\) but \(g(f(x))=x^{2}+2\). These are different functions, and their graphs have different vertices.

The two orders of composition give two different parabolas.

What is the domain of a composite function?

For \(f(g(x))\), an input \(x\) is allowed only if (1) \(x\) is in the domain of \(g\), and (2) \(g(x)\) is in the domain of \(f\).

Find the domain of the inner function \(g\).
Find which outputs of \(g\) the outer function \(f\) accepts.
Solve for the \(x\)-values that make \(g(x)\) one of those outputs.

Example: Let \(f(x)=\sqrt{x}\) and \(g(x)=x-4\). We need \(g(x)\ge 0\), so \(x-4\ge 0\) and \(x\ge 4\). The domain of \(f\circ g\) is \([4,\infty)\). Notice that \(g\) alone accepts every real number; it is the outer function \(f\) that cuts the domain down. Always check both stages, because a composite can have a smaller domain than either function alone.

Domain of the composite: all x from 4 rightward.

You can also run the process backward. To decompose \(h(x)=(5x-1)^{4}\), let the inner function be \(g(x)=5x-1\) and the outer function \(f(x)=x^{4}\).

Function operations lesson: questions and answers

What does f∘g mean?

It means the composite function f(g(x)). Apply g to x first, then apply f to that result. The small circle is read as ‘composed with.’

Is f(g(x)) the same as f(x) times g(x)?

No. f(g(x)) feeds the output of g into f. The product f(x)g(x) multiplies two separate outputs. For f(x) = x + 1 and g(x) = 2x, f(g(3)) = 7 but f(3)g(3) = 24.

How do you find the domain of a quotient of functions?

Find the domain shared by f and g, then remove every x where g(x) = 0. A zero denominator is never allowed.

How do you decompose a function?

Identify the inside expression and call it g(x), then write the remaining operation as f(x). For h(x) = sqrt(x^2 + 7), use g(x) = x^2 + 7 and f(x) = sqrt(x).

Can f(g(x)) equal g(f(x))?

Sometimes, such as when f and g are inverses of each other or both are f(x) = x + 3. In general they differ, so always check the order.

How to use this function operations lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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