Mastering Mathematics Smartly
by Wee Wen Shih

A unique self-help website that provides comprehensive coverage of mathematics at A-level & beyond, written in a student-friendly style.

Composite functions

Let us summarise some key points:

(1) For the composite function f g to exist, check that the range of g (from its graph) is a subset of or an equal set to the domain of f (usually given).

(2) The domain of f g is the domain of g.

(3) To determine the range of f g, find the range of g and then use the graph of f to obtain the range of y-values.

See example.

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Unusual composite functions

The usual composite functions (if they exist) have the forms \text{f} \circ \text{g} and/or \text{g} \circ \text{f}.

What about \text{f}^{-1} \circ \text{g} and/or \text{f} \circ \text{g}^{-1}? One should not be caught by the unusual forms but, instead, proceed with the standard process of checking for the existence of the composite function, i.e. range of the first function \subseteq domain of the second function.

So, in order to check for the existence of \text{f}^{-1} \circ \text{g}, we determine whether this holds: \text{R}_{\text{g}} \subseteq \text{D}_{\text{f}^{-1}}. Recall that \text{D}_{\text{f}^{-1}} = \text{R}_{\text{f}}.

If the composite function \text{f}^{-1} \circ \text{g} exists, then its domain is the domain of the first function as usual, i.e. \text{D}_{\text{g}}.

To find the range of \text{f}^{-1} \circ \text{g}, we will need to sketch the graph of y = \text{f}^{-1}(x); then put \text{R}_{\text{g}} on the x-axis (of the graph) to obtain the range.