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All courses Math III · F-BF.1.b 18 of 55
Combine studied function types arithmetically to build models.

Form a function sum and intersect the domains

Problem
Given f(x)=x² and g(x)=3x+5, form (f+g)(x), simplify without combining unlike terms, and enter the intersection of the two real domains. No contextual interpretation is requested because quantities and units are not supplied.
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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

Write the function sum

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

By definition, the sum of two functions adds their outputs for the same input \(x\).

02

Substitute the given formulas

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

Here \(f(x)=x^2\) and \(g(x)=3x+5\), so those expressions are added directly.

03

Simplify the combined expression

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

There are no like terms to combine further, so this is the finished expression for the sum.

04

Intersect the component domains

\[\text{domain}(f)∩\text{domain}(g)=(-∞,∞)\]

Both polynomial component functions are defined for every real input, so their domain intersection is all real numbers.

05

State the final answer

\[\text{unsimplified}~\text{sum}=(x^2)+(3x+5);~\text{simplified}~(f+g)(x)=x^2+3x+5;~\text{shared}~\text{real}~\text{domain}=(-∞,∞)\]

The sum is \(x^2+3x+5\), and it is defined for every real input.

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Another way

  1. You can check with a sample input such as \(x=2\): \(f(2)=4\) and \(g(2)=11\), so the sum is \(15\), which matches \(2^2+3(2)+5=15\).

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Common mistake

A common mistake is to multiply the functions or to add only part of \(g(x)\), but \((f+g)(x)\) means add the entire outputs \(x^2\) and \(3x+5\).