Functions can be added and multiplied; we can find the difference between two functions and we can define their quotient. In this chapter, we explain what these operations mean and what rules or restrictions apply.
Addition
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).
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.
Example 1
Let $f(x)=x^2$f(x)=x2 and $g(x)=2x+1$g(x)=2x+1. 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.
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. The three graphs are shown below.
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.
Example 2
Let $f(x)=\frac{3}{x}$f(x)=3x and $g(x)=2x-\frac{1}{3}$g(x)=2x−13.
The product function $(fg)(x)$(fg)(x) is given by $f(x)\cdot g(x)$f(x)·g(x) over the domain $R\text{\}\left\{0\right\}$R\{0}. The domain has to be restricted to the real numbers without zero because this is the domain of $f$f.
Hence, $(fg)(x)=\frac{3}{x}\cdot\left(2x-\frac{1}{3}\right)$(fg)(x)=3x·(2x−13) and so,
$(fg)(x)=6-\frac{1}{x}$(fg)(x)=6−1x
The graphs are shown below.
Difference
The function $(fg)(x)=6-\frac{1}{x}$(fg)(x)=6−1x in Example $2$2 can be seen in another way. We have the constant function $h(x)=6$h(x)=6 added to the constant function $r(x)=(-1)$r(x)=(−1) times the function $k(x)=\frac{1}{x}$k(x)=1x.
Multiplying by the constant $(-1)$(−1) is really just an example of function multiplication.
So, we have the combined sum and product $(fg)(x)=h(x)+r(x)k(x)$(fg)(x)=h(x)+r(x)k(x). A difference can always be thought of as a sum where one of the summands has been multiplied by $(-1)$(−1).
Compare the graph below with that in Example $2$2.
Quotient
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.