# Combinations of Functions

Let and $f$ and $g$ be functions with domains ${X}_{{1}}$ and ${X}_{{2}}$. Then the functions ${f{+}}{g{}}$, ${f{-}}{g{}}$, ${f{{g{}}}}$, and $\frac{{f}}{{g{}}}$ are defined as follows:

1. ${\left({f{+}}{g}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}+{g{{\left({x}\right)}}}$. Domain is intersection of domains ${X}_{{1}}$ and ${X}_{{2}}$: ${X}_{{1}}\cap{X}_{{2}}$.
2. ${\left({f{-}}{g}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}$. Domain is intersection of domains ${X}_{{1}}$ and ${X}_{{2}}$: ${X}_{{1}}\cap{X}_{{2}}$.
3. ${\left({f{{g}}}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}\cdot{g{{\left({x}\right)}}}$. Domain is intersection of domains ${X}_{{1}}$ and ${X}_{{2}}$: ${X}_{{1}}\cap{X}_{{2}}$.
4. ${\left(\frac{{f}}{{g}}\right)}{\left({x}\right)}=\frac{{{f{{\left({x}\right)}}}}}{{{g{{\left({x}\right)}}}}}$. Domain is intersection of domains ${X}_{{1}}$ and ${X}_{{2}}$: and such ${x}$ that ${g{{\left({x}\right)}}}\ne{0}$: ${\left\{{x}\in{X}_{{1}}\cap{X}_{{2}},{g{{\left({x}\right)}}}\ne{0}\right\}}$.

Example. If ${f{{\left({x}\right)}}}=\sqrt{{{x}-{2}}}$ and ${g{{\left({x}\right)}}}=\sqrt{{{9}-{{x}}^{{2}}}}$ find ${f{+}}{g{}}$, ${f{-}}{g{}}$, ${f{{g{}}}}$, and $\frac{{f}}{{g{}}}$.

Domain of ${f{{\left({x}\right)}}}$ is ${x}-{2}\ge{0}$ or interval ${\left[{2},\infty\right)}$. Domain of ${g{{\left({x}\right)}}}$ is ${9}-{{x}}^{{2}}\ge{0}$ or interval ${\left[-{3},{3}\right]}$.

So, the intersection of domains is ${\left[{2},\ \infty\right]}\cap{\left[-{3},\ {3}\right]}={\left[{2},\ {3}\right]}$.

Thus,

${\left({f{+}}{g}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}+{g{{\left({x}\right)}}}=\sqrt{{{x}-{2}}}+\sqrt{{{9}-{{x}}^{{2}}}}$ for ${2}\le{x}\le{3}$.

${\left({f{-}}{g}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}=\sqrt{{{x}-{2}}}-\sqrt{{{9}-{{x}}^{{2}}}}$ for ${2}\le{x}\le{3}$.

${\left({f{{g}}}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}{g{{\left({x}\right)}}}=\sqrt{{{x}-{2}}}\sqrt{{{9}-{{x}}^{{2}}}}=\sqrt{{{\left({x}-{2}\right)}{\left({9}-{{x}}^{{2}}\right)}}}$ for ${2}\le{x}\le{3}$.

${\left(\frac{{f}}{{g}}\right)}{\left({x}\right)}=\frac{{f{{\left({x}\right)}}}}{{g{{\left({x}\right)}}}}=\frac{\sqrt{{{x}-{2}}}}{\sqrt{{{9}-{{x}}^{{2}}}}}$ for ${2}\le{x}<{3}$.

Notice that the domain of $\frac{{f}}{{g{}}}$ is the interval $\left[2,3\right)$, because we must exclude the points where $g\left(x\right)=0$, i.e. $x=\pm 3$.