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Algebra / Graphing and Functions

Combining Functions: Adding, Subtracting, Multiplying, Dividing

Two functions can be combined with ordinary arithmetic to make a new one. (f + g)(x) simply means f(x) + g(x), and the same pattern covers subtraction, multiplication and division. The arithmetic is easy; the part worth care is the domain, because the combination is only defined where both original functions are — and division adds one more restriction on top. This lesson covers all four operations, their domains, and how the sum looks on a graph.

Practice Problems
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Once you have two functions, arithmetic gives you four more for free. The notation looks unfamiliar at first but the idea is as simple as it sounds: do both, then combine.

The four ways to combine two functions with arithmetic
The four arithmetic combinations

The Four Operations

The four definitions: sum, difference, product and quotient of two functions, with the non-zero condition on the quotient
Each combination evaluates both functions first, then combines.

The brackets on the left are doing something specific. is the name of a new function, and the after it is that function’s input. It is not multiplication.

Note the order matters for the two non-commutative operations: and differ by a sign, and is not .

Combining at a Single Input

The most straightforward case. Evaluate each function separately, then combine the two numbers.

Step-by-step evaluation of (f + g)(2) as f(2) + g(2) = 5 + 3 = 8
Two ordinary evaluations, then one addition.

With and :

You could instead build the combined formula first and then substitute — both give the same answer. Evaluating separately is usually less error-prone.

Combining as Formulas

To get a general expression, substitute the rules and simplify.

With the same and :

Keep the brackets when subtracting. The minus sign applies to every term of , so becomes . Dropping that bracket and writing is the single most common slip in this topic.

The Domain of a Combination

This is the part that earns marks. A combined function can only work where both originals work.

For the quotient there is one more condition:

The graph of g crossing the x-axis at x = 2, with a dashed vertical line marking the value the quotient must exclude
Every zero of g has to be removed from the domain of f/g.

Worked case. and .

  • Domain of : . Domain of : all reals.
  • Intersection: , which is the domain of , and .
  • For , also exclude , i.e. . Domain: .

A restriction from the original functions never disappears, even if the algebra tidies up. See Domain for finding each one.

Adding Functions on a Graph

Addition has a clean visual meaning: stack the heights.

Two lines f and g with vertical bars at x = 3 showing their heights stacked to give the height of f + g
At each input, the sum's height is the two heights added.

Pick an , measure how far each graph sits above the axis, add those two distances, and plot the result. Doing that at several inputs traces the sum curve. This is sometimes called addition of ordinates.

Signs work naturally: where is below the axis its height is negative, so it pulls the sum down. Where crosses zero, the sum touches .

Worked Example A: All Four at a Point

, . Find all four combinations at .

, .

Worked Example B: Subtraction With Brackets

, . Find .

Every sign inside flipped. Check one value: at , and , so ; the combined formula gives . ✓

Worked Example C: A Quotient Domain

, . Find and its domain.

Both domains are all reals, but at . Domain: .

Worked Example D: A Restriction That Survives

, . Find and its domain.

The simplified formula looks like it accepts anything, but the product was only ever defined where both roots were. Domain: .

This is the key idea — simplification does not restore a lost domain.

Worked Example E: A Real Combination

A stall’s revenue is and its costs are , for items.

Profit is the difference:

Break-even is , so — meaning 29 items, since you cannot sell a fraction of one.

Common Mistakes to Avoid

  • Dropping the bracket when subtracting. .
  • Confusing with . One multiplies outputs; the other is composition.
  • Forgetting to intersect the domains. Both functions must accept the input.
  • Forgetting the extra condition on . Zeros of are excluded.
  • Letting a domain restriction vanish on simplification. It survives the algebra.
  • Reading as multiplication. It is the name of the new function.
  • Assuming subtraction commutes. and are negatives of each other.

Where This Leads Next

  • Function composition — the other way to build a function from two, and a genuinely different operation.
  • Profit, revenue and cost models, where profit is literally a difference of functions.
  • Rational functions, which are quotients of polynomials.
  • Partial fractions, splitting a quotient back into a sum.
  • Calculus, where the sum, product and quotient rules differentiate exactly these combinations.

Practice Problems

Use and for Problems 1–5.

Problem 1. Find .

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, , so .

Problem 2. Find .

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.

Problem 3. Find .

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.

Problem 4. Find as a formula.

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.

Problem 5. Find the domain of .

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at , so all reals except 0.

Problem 6. For and , find .

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.

Problem 7. For and , state the domain of .

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Intersection of and all reals: .

Problem 8. For and , simplify and give its domain.

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, but is still excluded. Domain: all reals except 1.

Problem 9. Revenue is and cost is . Write the profit function.

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.

Problem 10. At , has height 5 and has height on a graph. What is the height of there?

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. The negative height pulls the sum down.

Quick Reference

CombinationDefinitionDomain
intersection of domains
intersection of domains
intersection of domains
intersection, minus zeros of
Subtractingbracket the whole of
On a graphadd the heights at each
vs product vs compositiondifferent operations

Arithmetic combination is one of two ways to build a new function from two old ones; the other is composition. Both rely on function notation and on working out each domain. More Algebra lessons are available.

Frequently Asked Questions

What does (f + g)(x) mean?+

It means — evaluate both functions at the same input and add the results. The notation is shorthand for a new function built from two existing ones.

How do you add two functions?+

Add their outputs at each input. If and , then . At a specific input, evaluate each separately and add: .

What is the domain of a combined function?+

The intersection of the two domains — every value that both functions accept. For the quotient you must also exclude any input where , since that would divide by zero.

Why does f/g have an extra restriction?+

Because division by zero is undefined. Even if is in both domains, if at that input then has no value, so that input must be removed from the domain of the quotient.

Is (fg)(x) the same as f(g(x))?+

No, and confusing them is a common error. multiplies the two outputs, . is composition — it feeds the output of into as an input.

Can you combine functions on a graph?+

Yes. To sketch , pick an -value, measure the height of each graph there, add the heights, and plot the total. Repeating that at several inputs traces out the sum.

Related lessons

Function Composition: Feeding One Function Into AnotherFunctions: The One-Output Rule and the Vertical Line TestFunction Notation: Reading and Using f(x)Domain of a Function: Which Inputs Are AllowedPiecewise Functions: Different Rules on Different IntervalsRange of a Function: Which Outputs Are Possible