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All courses Math II · F-BF.1.b 15 of 73
Combine standard function types arithmetically to create models.

Form and interpret the sum of two functions, (f+g)(x)

Problem
Add functions f(x)=2x+5 and g(x)=x²-3 to model a combined quantity.
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Hint

Write \((f+g)(x)\) as \(f(x)+g(x)\), then substitute the two formulas and combine like terms.

Adding functions means adding their outputs for the same input value.

Solution walkthrough

01

Write the sum definition

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

To add two functions, add the expressions that define their outputs.

02

Substitute the given functions

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

Replace \(f(x)\) with \(2x+5\) and \(g(x)\) with \(x^2-3\).

03

Combine like terms

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

The constant terms \(5\) and \(-3\) combine to make \(2\).

04

State the combined function

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

That is the function modeling the combined quantity.

+

Another way

  1. You can remove the parentheses first and then reorder terms as \(x^2+2x+5-3\) before simplifying.

!

Common mistake

Multiplying the two functions instead of adding them, even though the notation is \((f+g)\).