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

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

Worked example

Question 1

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

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

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

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

 

Multiplication

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

Worked example

Question 2

Let $f(x)=\frac{3}{x}$f(x)=3x​ and $g(x)=2x-\frac{1}{3}$g(x)=2x−13​.

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

Think: The product function $(fg)(x)$(fg)(x) is given by $f(x)\cdot g(x)$f(x)·g(x) over the domain $\left\{R|x\ne0\right\}${R|x≠0}.  The domain has to be restricted to the real numbers without zero because this is the domain of $f$f.

Do: Therefore, $(fg)(x)=\frac{3}{x}\cdot\left(2x-\frac{1}{3}\right)$(fg)(x)=3x​·(2x−13​) and so,

$(fg)(x)=6-\frac{1}{x}$(fg)(x)=6−1x​

 

Division

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

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

Worked example

Question 3

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

Find $(\frac{f}{g})(x)$(fg​)(x) and its domain.

Think: The quotient function is given by $(\frac{f}{g})(x)=\frac{f(x)}{g(x)}$(fg​)(x)=f(x)g(x)​, provided that $g(x)\ne0$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, $(\frac{f}{g})(x)=\frac{x^2+5x+6}{2x+1}$(fg​)(x)=x2+5x+62x+1​, provided that $x\ne\frac{1}{2}$x≠12​.

 

Practice questions

Question 4

Question 5

Question 6