Sum, difference, product and division
Sum
In the set of real functions of real variable we can define different operations.
The function sum $f + g$ is a function that assigns to every real number $x$ the sum of the images of the function $f$ and of the function $g$: $$(f+g)(x)=f(x)+g(x)$$
The sum function is defined when $x$ belongs simultaneously to the domain of $f$ and of $g$: $$Dom(f+g)=Dom(f)\cap Dom(g)$$
Consider the functions $\displaystyle f(x)=\frac{1}{x}$ and $g(x)=x-2$ and compute $(f + g) (x)$. $$(f+g)(x)=f(x)+g(x)=\displaystyle \frac{1}{x}+x-2=\frac{x^2-2x+1}{x}$$Therefore,$$\displaystyle (f+g)(x)=\frac{x^2-2x+1}{x}$$
Difference
The subtraction function (or difference) $f-g$ is a function that assigns to every real number $x$ the difference of the images of the function $f$ and of the function $g$. $$(f-g)(x)=f(x)-g(x)$$
The function differerence is defined when $x$ belongs simultaneously to the domain of $f$ and of $g$: $$Dom(f-g)=Dom(f)\cup Dom(g)$$
Consider the functions $f(x)=\displaystyle \frac{1}{x}$ and it $g(x)=x-2$ and compute $(f - g) (x)$.
$$(f-g)(x)=f(x)-g(x)=\displaystyle \frac{1}{x}-(x-2)=\frac{-x^2+2x+1}{x}$$Therefore,$$(f-g)(x)=\displaystyle \frac{-x^2+2x+1}{x}$$
Product
The function product $f \cdot g$ is a function that assigns to every real number $x$ the product of the images of the function $f$ and of the function $g$. $$(f\cdot g)(x)=f(x)\cdot g(x)$$
The function product is defined when $x$ belongs simultaneously to the domain of $f$ and of $g$: $$Dom(f\cdot g)=Dom(f) \cap Dom(g)$$
Consider the functions $\displaystyle f(x)=\frac{1}{x}$ and it $g(x)=x-2$ and compute $(f \cdot g) (x)$.
$$(f \cdot g)(x)=f(x)\cdot g(x)=\frac{1}{x}\cdot (x-2)=\frac{x-2}{x}$$Therefore,$$\displaystyle (f \cdot g)(x)=\frac{x-2}{x}$$
Division
The function division $\displaystyle \frac{f}{g}$ is a function that assigns to every real number $x$ the division of the images of the function $f$ and of the function $g$. $$\displaystyle \Big(\frac{f}{g}\Big)(x)=\frac{f(x)}{g(x)}$$
The function quotient is defined when $x$ belongs simultaneously to the domain of $f$ and of $g$, and we also have that $g(x)\neq 0$. That is: $$\displaystyle Dom\Big(\frac{f}{g}\Big)=Dom(f) \cap Dom(g)-\{x \in \mathbb{R} \mid g(x)=0\}$$
Consider the functions$f(x)=x^2+3$ and $g(x)=x^2+1$ compute $\displaystyle \Big(\frac{f}{g}(x)\Big)$:
$$\displaystyle \Big(\frac{f}{g}\Big)(x)=\frac{f(x)}{g(x)}=\frac{x^2+3}{x^2+1}$$ Therefore, $$\displaystyle \Big(\frac{f}{g}\Big)(x)=\frac{x^2+3}{x^2+1}$$