If $f$ and $g$ are two functions, we can define new functions $f+g$,$f-g$, $f.g$, and $f/g$ by the formulas:

\begin{align*}

(f+g)(x) & =f(x)+g(x)\\

(f-g)(x) & =f(x)-g(x)\\

(f.g)(x)= & f(x).g(x)\\

(f/g)(x) & =\frac{f(x)}{g(x)}

\end{align*}

The domains of $f+g$, $f-g$, $f.g$ consist of all $x$ for which both $f(x)$ and $g(x)$ are defined:

\[

\{x|\ x\in Dom(f)\text{ and }x\in Dom(g)\}

\]

\[

Dom(f)\cap Dom(g)

\]

The domain of $f/g$ is trickier. Its domain consists of all $x$ for which $f(x)$ and $g(x)$ are defined but $g(x)\neq0$, because division by zero is not defined. Therefore:

\[ \bbox[#F2F2F2,5px,border:2px solid black]{\large Dom(f/g)=\{x|\ x\in Dom(f)\text{ and }x\in Dom(g)\text{ and }g(x)\neq0\},}\]
which also can be written as:

\[

Dom(f/g)=Dom(f)\cap Dom(g)\cap\{x|\ g(x)\neq0\}.

\]

## Graphs of Combined Functions

Suppose the graphs of $f$ and $g$ are known:

- to obtain the graph of the function $f+g$, we just add the corresponding $y$-coordinates $f(x)$ and $g(x)$ at each $x$ that belongs both to the domain of $f$ and the domain of $g$
- to obtain the graph of the function $f+g$, we just subtract the $y$-coordinate $g(x)$ from the corresponding $y$-coordinate $f(x)$ at each $x$ that belongs both to the domain of $f$ and the domain of $g$
- to obtain the graph of the function $fg$, we multiply the corresponding$y$-coordinates $f(x)$ and $g(x)$ at each $x$ that belongs both to the domain of $f$ and the domain of $g$.