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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. $$$x=\pm 3$$$.