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Linear Relationships

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, and 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 and 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, and 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 and 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.
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Exercise

Given the linear functions and find the function

Show Solution
Solution
We add functions by combining like terms. Since begins with a negative term, we can put parentheses around it before adding.
The new function, can be written as
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Exercise

The functions and are given as and Find if

Show Solution
Solution
To begin, we can write the rule of by multiplying and
Next, to find we'll substitute into the rule of and simplify.
Thus,
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