# 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, $$\displaystyle \Big(\frac{f}{g}\Big)(x)=\frac{x^2+3}{x^2+1}$$$