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All courses Algebra I · F-BF.1.b 61 of 76
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 \(g(x)=x^2-3\) 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

Use the definition of function sum

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

A function sum evaluates both functions at the same input x and adds their outputs.

02

Substitute both rules with grouping

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

Replace f(x) and g(x) by their complete expressions. Parentheses preserve the signed constant -3.

03

Combine like terms

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

The quadratic and linear terms are unlike and remain separate. Only the constants 5 and -3 combine to 2.

04

Check with one input

\[\begin{aligned} x=0:~f(0)+g(0)=5+(-3)=2 \\ \text{combined}:~0+0+2=2 \end{aligned}\]

Both the original output sum and simplified rule give 2 at x=0, confirming the combined function.

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Another way

  1. Arrange the function outputs by degree, then add coefficient positions.

!

Common mistake

Do not multiply the functions. The notation f+g specifies addition of outputs at the same input.

Solution walkthrough video