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Functions can be combined, through addition, subtract, multiplication, and division to create a new function. The resulting function can have similar or new characteristics, depending on the original functions and which *function operation* is used.

Functions behave similar to numbers when they are combined to create new functions. Common ways of combining functions include addition, subtraction, multiplication and division. ### Method

### Addition and Subtraction

When adding or subtracting functions, it is possible to find the resulting function by combining like terms. If two functions, $f(x)$ and $g(x),$ are combined through addition or subtraction and one or both have a restricted domain, the domain of the resulting function is restricted to interval(s) where both $f(x)$ and $g(x)$ are defined. ### Method

### Multiplication

When multiplying two functions, the Distributive Property can be used. Meaning, every term in one function is multiplied with every term in the other. If two functions, $f(x)$ and $g(x),$ are combined through multiplication and one or both have a restricted domain, the domain of the resulting function is restricted to interval(s) where both $f(x)$ and $g(x)$ are defined. ### Method

### Division

When two functions are combined through division it is necessary to take into account that division by zero is undefined. As a consequence the domain of the resulting function is restricted to interval(s) where both functions are defined and where the denominator does not equal zero.

Given the linear functions $f(x)=3x+5$ and $g(x)=-x+2,$ find the function $h(x)=f(x)+g(x).$

Show Solution

We add functions by combining like terms. Since $g(x)$ begins with a negative term, we can put parentheses around it before adding.
The new function, $h(x),$ can be written as $h(x)=2x+7.$

$h(x)=f(x)+g(x)$

SubstituteII

$f(x)=3x+5$, $g(x)=-x+2$

$h(x)=3x+5+(-x+2)$

RemovePar

Remove parentheses

$h(x)=3x+5−x+2$

CommutativePropAdd

Commutative Property of Addition

$h(x)=3x−x+5+2$

SimpTerms

Simplify terms

$h(x)=2x+7$

The functions $f$ and $g$ are given as $f(x)=x−4$ and $g(x)=2x−1.$ Find $h(2)$ if $h(x)=f(x)⋅g(x).$

Show Solution

To begin, we can write the rule of $h(x)$ by multiplying $f$ and $g.$
Next, to find $h(2),$ we'll substitute $x=2$ into the rule of $h$ and simplify.
Thus, $h(2)=-6.$

$h(x)=f(x)⋅g(x)$

SubstituteII

$f(x)=x−4$, $g(x)=2x−1$

$h(x)=(x−4)(2x−1)$

MultPar

Multiply parentheses

$h(x)=2x_{2}−x−8x+4$

SimpTerms

Simplify terms

$h(x)=2x_{2}−9x+4$

$h(x)=2x_{2}−9x+4$

Substitute

$x=2$

$h(2)=2⋅2_{2}−9⋅2+4$

CalcPow

Calculate power

$h(2)=2⋅4−9⋅2+4$

Multiply

Multiply

$h(2)=8−18+4$

AddSubTerms

Add and subtract terms

$h(2)=-6$

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