Applying two functions in order

The composite \(gf(x)=g(f(x))\) means apply f first, then g to its result. Substitute the entire expression for the inner output into every occurrence of the outer function’s input.

Example

If \(f(x)=2x+1\) and \(g(x)=x^2\), then \(gf(x)=(2x+1)^2\), while \(fg(x)=2x^2+1\). At \(x=2\), these give 25 and 9 respectively; the order matters.

For \(h(x)=\dfrac3{x+2}\), \(hg(x)=\dfrac3{x^2+2}\). Use brackets so substitution changes the whole input, not just one term.

Evaluate the inner function first, then check that its output is a permitted input for the outer function. Brackets preserve this order.