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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.

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.

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.

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

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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 $h(x)=2x+7.$

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

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

RemoveParRemove parentheses

$h(x)=3x+5−x+2$

CommutativePropAddCommutative Property of Addition

$h(x)=3x−x+5+2$

SimpTermsSimplify 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 $h(x)=f(x)⋅g(x).$

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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, $h(2)=-6.$

$h(x)=f(x)⋅g(x)$

$h(x)=(x−4)(2x−1)$

MultParMultiply parentheses

$h(x)=2x_{2}−x−8x+4$

SimpTermsSimplify terms

$h(x)=2x_{2}−9x+4$

$h(x)=2x_{2}−9x+4$

Substitute$x=2$

$h(2)=2⋅2_{2}−9⋅2+4$

CalcPowCalculate power

$h(2)=2⋅4−9⋅2+4$

MultiplyMultiply

$h(2)=8−18+4$

AddSubTermsAdd and subtract terms

$h(2)=-6$

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