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All courses Algebra II · F-BF.1.b 25 of 55
Combine studied function types arithmetically to build models.

Form a function sum and intersect the domains

Problem
For \(f(x)~=~x^2\) and \(g(x)~=~3x~+~5\), find \((f~+~g)(x)\) and its shared domain.
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Hint

Substitute both formulas with parentheses, then identify restrictions from radicals, logarithms, and denominators.

(f+g)(x)=f(x)+g(x), and its domain is domain(f)∩domain(g).

Solution walkthrough

01

Define pointwise addition

\[(f+g)(x)=f(x)+g(x)\]

A function sum adds the two outputs at the same allowed input x.

02

Substitute and simplify

\[(f+g)(x)=x^2+(3x+5)=x^2+3x+5\]

Insert the complete formulas for f and g, then remove the grouping without changing signs.

03

Intersect the domains

\[\text{Sum}:~(f+g)(x)=x^2+3x+5.~\text{Shared}~\text{domain}:~\text{all}~\text{real}~\text{numbers}.\]

Both polynomial inputs accept every real x, so their domain intersection is all real numbers, choice A.

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Another way

  1. Evaluate the two polynomial formulas in parallel and combine like terms only after the pointwise sum is written.

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Common mistake

A function sum combines outputs f(x) and g(x); it does not add their input variables separately.

Solution walkthrough video