(1)  Sum of \(f\) and \(g: (f + g)(x) = f(x) + g(x)\) 

(2)  Difference of \(f\) and \(g:(f-g)(x)=f(x)-g(x)\) 

(3)  Product of f and \(g:  (f . g)(x) = f(x) . g(x)\) 

Example 7: Let \(f(x) = 2x + 3\) and \(g(x) = x - 2\), find the value of  \((f + g)(x)\) and  \((f - g)(x)\)

Solution: \((f + g)(x) = f(x) + g(x) = (2x + 3) + (x-2)\)     \(= 3x +1, (f - g)(x) = f(x) - g(x) = (2x + 3) - (x - 2)\) 

\(= x + 5\)

Example 8: Let \(f(x) = 2x + 3\) and \(g(x) = x - 2\), find the value of  \(f(g(x))\) and  \(g(f(x))\)

Solution: \(f(g(x)) = f(x - 2) = 2(x - 2) + 3 = 2x -1\)

\(g(f(x))\) = \(g(2x + 3) = (2x + 3) - 2 = 2x +1\)

Hence it is seen that \(f(g(x))\) is not equal to \(g(f(x))\)

Example 9: If \(f(x) = x + 2\) and \(g\left( x \right){\rm{ }} = {\rm{ }}{x^2}\), find \(f(g(x))\) and \(g(f(x))\)

Solution: \(f\left( {g\left( x \right)} \right){\rm{ }} = f\left( {{x^2}} \right){\rm{ }} = {x^2} + {\rm{ }}2\).  The function \(g(x)\) squares any number.  The function \(f(x)\) simply adds two to any number.  \(g\left( {f\left( x \right)} \right){\rm{ }} = g\left( {x + {\rm{ }}2} \right){\rm{ }} = {\rm{ }}{\left( {x + 2} \right)^2}\)   This time, \(f(x)\) is applied first and then \(g(x)\). 

Notice that the answers are not the same illustrating that the composite function is not commutative.  This means the order in which the problem is written is important. 

Example 10: If \(f\left( {x + \frac{1}{x}} \right) = {x^2} + \frac{1}{{{x^2}}}\), find the value of \(f(20)\)

Solution: Suppose \(x + \frac{1}{x} = t\) 

Squaring both sides, we get 

\({x^2} + \frac{1}{{{x^2}}} + 2 = {t^2}\,\,or\,\,{x^2} + \frac{1}{{{x^2}}}\, = {t^2} - 2\,\,\)

\( \Rightarrow \) \(f\left( t \right){\rm{ }} = {t^2}-{\rm{ }}2\)

or \(f\left( {20} \right){\rm{ }} = {\rm{ }}{20^2}-{\rm{ }}2{\rm{ }} = {\rm{ }}398\).