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Combinations of Functions

Let and ff and gg be functions with domains X1{X}_{{1}} and X2{X}_{{2}}. Then the functions f+g{f{+}}{g{}}, f−g{f{-}}{g{}}, fg{f{{g{}}}}, and fg\frac{{f}}{{g{}}} are defined as follows:

  1. (f+g)(x)=f(x)+g(x){\left({f{+}}{g}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}+{g{{\left({x}\right)}}}. Domain is intersection of domains X1{X}_{{1}} and X2{X}_{{2}}: X1∩X2{X}_{{1}}\cap{X}_{{2}}.
  2. (f−g)(x)=f(x)−g(x){\left({f{-}}{g}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}. Domain is intersection of domains X1{X}_{{1}} and X2{X}_{{2}}: X1∩X2{X}_{{1}}\cap{X}_{{2}}.
  3. (fg)(x)=f(x)⋅g(x){\left({f{{g}}}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}\cdot{g{{\left({x}\right)}}}. Domain is intersection of domains X1{X}_{{1}} and X2{X}_{{2}}: X1∩X2{X}_{{1}}\cap{X}_{{2}}.
  4. (fg)(x)=f(x)g(x){\left(\frac{{f}}{{g}}\right)}{\left({x}\right)}=\frac{{{f{{\left({x}\right)}}}}}{{{g{{\left({x}\right)}}}}}. Domain is intersection of domains X1{X}_{{1}} and X2{X}_{{2}}: and such x{x} that g(x)≠0{g{{\left({x}\right)}}}\ne{0}: {x∈X1∩X2,g(x)≠0}{\left\{{x}\in{X}_{{1}}\cap{X}_{{2}},{g{{\left({x}\right)}}}\ne{0}\right\}}.

Example. If f(x)=x−2{f{{\left({x}\right)}}}=\sqrt{{{x}-{2}}} and g(x)=9−x2{g{{\left({x}\right)}}}=\sqrt{{{9}-{{x}}^{{2}}}} find f+g{f{+}}{g{}}, f−g{f{-}}{g{}}, fg{f{{g{}}}}, and fg\frac{{f}}{{g{}}}.

Domain of f(x){f{{\left({x}\right)}}} is x−2≥0{x}-{2}\ge{0} or interval [2,∞){\left[{2},\infty\right)}. Domain of g(x){g{{\left({x}\right)}}} is 9−x2≥0{9}-{{x}}^{{2}}\ge{0} or interval [−3,3]{\left[-{3},{3}\right]}.

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

Thus,

(f+g)(x)=f(x)+g(x)=x−2+9−x2{\left({f{+}}{g}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}+{g{{\left({x}\right)}}}=\sqrt{{{x}-{2}}}+\sqrt{{{9}-{{x}}^{{2}}}} for 2≤x≤3{2}\le{x}\le{3}.

(f−g)(x)=f(x)−g(x)=x−2−9−x2{\left({f{-}}{g}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}-{g{{\left({x}\right)}}}=\sqrt{{{x}-{2}}}-\sqrt{{{9}-{{x}}^{{2}}}} for 2≤x≤3{2}\le{x}\le{3}.

(fg)(x)=f(x)g(x)=x−29−x2=(x−2)(9−x2){\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≤x≤3{2}\le{x}\le{3}.

(fg)(x)=f(x)g(x)=x−29−x2{\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≤x<3{2}\le{x}<{3}.

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