Operations on Functions

Q: What does $(f + g)(x)$ mean?

Add the outputs: $(f + g)(x) = f(x) + g(x)$. Substitute both rules and combine.

Q: How is this different from composition?

These operations combine outputs with $+, -, \times, \div$. Composition, $(f \circ g)(x)$, feeds one function's output into the other — a different operation.

You can combine two functions the way you combine numbers: add, subtract, multiply, or divide their outputs. $(f + g)(x)$ just means $f(x) + g(x)$ .

Each operation applies the two rules and combines the results. Division adds one caveat — the denominator function cannot be zero.

The four operations

$(f + g)(x) = f(x) + g(x)$ , and subtraction, multiplication, and division follow the same pattern. Substitute each function's rule, then combine like terms.

For $(f - g)(x)$ , distribute the subtraction across all of $g$ 's terms — the same sign trap as subtracting polynomials.

Domain of a combination

The combined function is defined only where both original functions are defined. For a quotient $\displaystyle (\frac{f}{g})(x)$ , also exclude any x that makes $g(x) = 0$ .

So start from the overlap of the two domains and remove denominator zeros for division.

Worked examples

Example 1: a difference of functions

For $f(x) = x^2 - 4x$ and $g(x) = 3x + 2$ , find $(f - g)(x)$ .

Subtract g from f

(x^2 - 4x) - (3x + 2)

Distribute the minus

x^2 - 4x - 3x - 2

Combine like terms

x^2 - 7x - 2

Answer: $x^2 - 7x - 2$

Example 2: a product

For $f(x) = x$ and $g(x) = x + 1$ , find $(fg)(x)$ .

Multiply the rules

x(x + 1)

Distribute

x^2 + x

Answer: $x^2 + x$

Example 3: a quotient and its excluded value

For $f(x) = x^2 - 9$ and $g(x) = x - 3$ , find $\displaystyle (\frac{f}{g})(x)$ and state its domain.

Divide the rules

\dfrac{x^2 - 9}{x - 3}

Factor the numerator

\dfrac{(x + 3)(x - 3)}{x - 3}

Cancel the common factor

x + 3

The denominator function is zero at 3, so remove it

x \neq 3

Answer: $x + 3$ , for all $x \neq 3$

Try one yourself

Common questions

What does

(f + g)(x)

mean?

Add the outputs: $(f + g)(x) = f(x) + g(x)$ . Substitute both rules and combine.

What is the domain of a combined function?

Where both functions are defined. For division, also remove any x that makes the denominator function zero.

How is this different from composition?

These operations combine outputs with $+, -, \times, \div$ . Composition, $(f \circ g)(x)$ , feeds one function's output into the other — a different operation.

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Operations on Functions

The four operations

Domain of a combination

Worked examples

Example 1: a difference of functions

Example 2: a product

Example 3: a quotient and its excluded value

Try one yourself

Common questions

Keep learning

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Function Notation, Evaluating & Interpreting

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