Composite functions

Given \(f(x) = 3x + 2\), we are often asked to find \(f(2)\) or \(f( - 3)\). To do this we substitute \(2\) or \(- 3\) for \(x\). So, \(f(2) = 3(2) + 2 = 8\) and \(f( - 3) = 3( - 3) + 2 = - 7\).

Sometimes, however, we are asked to find the result of a function of a function. That is, replacing \(x\) in the example above with another function.

Follow this worked example:

\[f(x) = 10x + 7\]

\[g(x) = 3x\]

Find \(f(g(x))\)

Replace \(x\) with the function

\[f(g(x)) = 10(g(x)) + 7\]

\[f(3x) = 10(3x) + 7\]

\[f(g(x)) = 30x + 7\]

Question

\[f(x)=x+1\]

\[g(x) = 4{x^2} + 8x - 7\]

Find \(g(f(x))\)

\[g(f(x)) = 4{(f(x))^2} + 8(f(x)) - 7\]

Simplify:

\[g(x + 1) = 4{(x + 1)^2} + 8(x + 1) - 7\]

\[g(x + 1) = 4({x^2} + 2x + 1) + 8x + 8 - 7\]

\[g(f(x)) = 4{x^2} + 8x + 4 + 8x + 1\]

\[g(f(x)) = 4{x^2} + 16x + 5\]

curriculum-key-fact
When you're asked to find \(f(g(x))\) and \(g(f(x))\), the order is important. Apart from a few special cases, \(f(g(x))\) does not equal \(g(f(x))\).