Combining Linear Functions

Concept

Function Operation

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.

Method

Combining Functions

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.

Add the linear functions

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Exercise

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

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Solution

We add functions by combining like terms. Since

g (x)

begins with a negative term, we can put parentheses around it before adding.

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

SubstituteII

f (x) = 3 x + 5

,

g (x) = - x + 2

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

RemoveParRemove parentheses

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

CommutativePropAddCommutative Property of Addition

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

SimpTermsSimplify terms

h (x) = 2 x + 7

The new function,

h (x),

can be written as

h (x) = 2 x + 7 .

Multiply the linear functions

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Exercise

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

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Solution

To begin, we can write the rule of

h (x)

by multiplying

f

and

g .

h (x) = f (x) \cdot g (x)

SubstituteII

f (x) = x - 4

,

g (x) = 2 x - 1

h (x) = (x - 4) (2 x - 1)

MultParMultiply parentheses

h (x) = 2 x^{2} - x - 8 x + 4

SimpTermsSimplify terms

h (x) = 2 x^{2} - 9 x + 4

Next, to find

h (2),

we'll substitute

x = 2

into the rule of

h

and simplify.

h (x) = 2 x^{2} - 9 x + 4

Substitute

x = 2

h (2) = 2 \cdot 2^{2} - 9 \cdot 2 + 4

CalcPowCalculate power

h (2) = 2 \cdot 4 - 9 \cdot 2 + 4

MultiplyMultiply

h (2) = 8 - 18 + 4

AddSubTermsAdd and subtract terms

h (2) = - 6

Thus,

h (2) = - 6 .

Combining Linear Functions

Function Operation

Combining Functions

Addition and Subtraction

Multiplication

Division

Example

Add the linear functions

Example

Multiply the linear functions