BetterGrades Precalculus · Unit 3 · Lesson

Sums and differences of functions

Add and subtract functions pointwise, determine the common domain, and interpret combined outputs.

Opening

Start with the situation

Function sums and differences combine outputs at the same input.

Transformation language lets you read a complicated graph as a modified parent rather than a collection of disconnected points. That makes prediction possible before any calculator window is opened.

Before you begin

Prerequisite check

Recognize the parent function family.
Read domain, range, and key points.
Use coordinate mappings.

Core explanation

Explanation

Compute $f(x)$ and g(x), add or subtract them pointwise, and intersect the original domains.

Units must be compatible, and subtraction order matters.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through pointwise addition, domain intersection, or another equivalent representation.

Conceptual reading

What the idea is really doing

Transformations become reliable when they are treated as coordinate mappings. Outside operations change outputs; inside operations change the inputs that produce those outputs, which is why horizontal changes often appear to work in the opposite direction.

This lesson narrows that lens to one goal: add and subtract functions pointwise, determine the common domain, and interpret combined outputs. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Compute $f(x)$ .
$G(x)$ .
Add or subtract them pointwise.
Intersect the original domains.

Verification: Track at least one landmark point from the parent graph to the transformed graph, then verify the new domain, range, intercepts, or asymptotes from the formula.

Foundation walkthrough

Plan before calculating

Problem

Revenue $30q$ and cost $500+18q$ .

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Compute $f(x)$ and g(x), add or subtract them pointwise, and intersect the original domains.
Conclusion: Profit $12q-500$ .
Why the check works: A difference function represents net value.

Worked examples

See the idea in three forms

foundation example

Revenue $30q$ and cost $500+18q$ .

SolutionProfit $12q-500$ .

A difference function represents net value.

representation example

If $f(2)=5,g(2)=-1$ .

Solution $4$

This example expresses sums and differences of functions in a second form.

transfer example

Domain $sqrt(x-3)+ln x$ .

Solution $[3,∞)$

Units must be compatible, and subtraction order matters.

Pointwise addition. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: A difference function represents net value. — **Anchor figure · Pointwise addition**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: A difference function represents net value.

Domain intersection. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for sums and differences of functions. — **Mechanism figure · Domain intersection**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for sums and differences of functions.

Revenue-cost-profit gap. Compare the valid path with the tempting shortcut. The figure shows why confusing f(x)+g(x) with composition f(x+g(x)) leads to a false conclusion. — **Comparison and error figure · Revenue-cost-profit gap**
Compare the valid path with the tempting shortcut. The figure shows why confusing $f(x)+g(x)$ with composition $f(x+g(x))$ leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is confusing $f(x)+g(x)$ with composition $f(x+g(x))$ .

Check yourself

$f=2x,g=x^2$ ; $f+g$ .

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Attempt once to unlock the answer

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Practice

Ten concrete questions

Practice 101

$f=2x,g=x^2$ ; $f+g$ .

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Attempt once to unlock the answer

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Practice 202

If $f(2)=5,g(2)=-1$ .

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Attempt once to unlock the answer

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Practice 303

Domain $sqrt(x-3)+ln x$ .

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Practice 404

If $g-f=7,$ f-g.

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Practice 505

Explain why this conclusion is valid: Profit $12q-500$ . Use the foundation problem as evidence: Revenue $30q$ and cost $500+18q$ .

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Practice 606

Solve the representation example, then name the feature of sums and differences of functions that it illustrates: If $f(2)=5,g(2)=-1$

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is confusing $f(x)+g(x)$ with composition $f(x+g(x))$ .

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Practice 808

Connect two representations for this example: Revenue $30q$ and cost $500+18q$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Domain $sqrt(x-3)+ln x$ . Predict the effect, solve your new example, and compare it with the original.

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Practice 1010

Write a short verification checklist for sums and differences of functions, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Products and quotients of functions, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Lippman and Rasmussen, Precalculus Volume 1
Utah College Algebra
Stitz and Zeager, Precalculus

No long source passage is reproduced.

Revenue 30q30q30q and cost 500+18q500+18q500+18q.

Revenue 30q30q30q and cost 500+18q500+18q500+18q.

If f(2)=5,g(2)=−1f(2)=5,g(2)=-1f(2)=5,g(2)=−1.

Domain sqrt(x−3)+lnxsqrt(x-3)+ln xsqrt(x−3)+lnx.

Original BetterGrades manuscript, rights-separated references.

Revenue $30q$ and cost $500+18q$ .

Revenue $30q$ and cost $500+18q$ .

If $f(2)=5,g(2)=-1$ .

Domain $sqrt(x-3)+ln x$ .