Determining composite and inverse functionsComposite functions

Composite and inverse functions can be determined for trigonometric, logarithmic, exponential or algebraic functions.

Part ofMathsAlgebraic and trigonometric skills

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 \(g(f(x))\)