11.09 Operations with functions

Just as we can perform operations on polynomials, so too can we perform operations on different functions - adding, subtracting, multiplying or dividing them, provided we follow specific rules. The table below defines each of the four operations for functions.

 

Addition and subtraction

Two functions $$f and $$g may be combined as a sum $$f+g, meaning that for each $$x in the common domain we add the function values $$f(x) and $$g(x) to get $$(f+g)(x). The same can be done with the difference of two functions.

Note that this operation makes no sense unless $$x belongs to the domains of both $$f and $$g. It may be necessary to restrict the domain of one or both functions to meet this requirement.

Worked example

Question 1

Let $$f(x)=x2 and $$g(x)=2x+1.

Find $$(f+g)(x) and its domain.

Think: The domains of both functions are the real numbers. So, the sum function $$(f+g)(x) will also have the real numbers for its domain.

Do: We have, $$(f+g)(x)=f(x)+g(x) for each $$x in the domain. Therefore, $$(f+g)(x)=x2+2x+1, and Its domain is the set of real numbers.

 

Multiplication

We combine functions $$f and $$g as a product $$fg by defining $$(fg)(x)=f(x)·g(x) for each $$x in the common domain of $$f and $$g.

Worked example

Question 2

Let $$f(x)=3x​ and $$g(x)=2x−13​.

Find $$(fg)(x) and its domain.

Think: The product function $$(fg)(x) is given by $$f(x)·g(x) over the domain $${R|x≠0}.  The domain has to be restricted to the real numbers without zero because this is the domain of $$f.

Do: Therefore, $$(fg)(x)=3x​·(2x−13​) and so,

$$(fg)(x)=6−1x​

 

Division

We can define a quotient function $$h(x)=f(x)g(x)​ in a similar way to the way we defined the other operations, provided the domains of $$f and $$g are the same and we do not include in the domain values of $$x that make $$g(x)=0.

Such functions are called rational functions when $$f and $$g are both polynomials.

Worked example

Question 3

Given $$f(x)=x2+5x+6 and $$g(x)=2x+1,

Find $$(fg​)(x) and its domain.

Think: The quotient function is given by $$(fg​)(x)=f(x)g(x)​, provided that $$g(x)≠0. First, we must find the quotient. Then determine restrictions on its domain by finding the values that make the denominator zero.

Do: Therefore, $$(fg​)(x)=x2+5x+62x+1​, provided that $$x≠12​.

 

Practice questions

Question 4

Question 5

Question 6