Composite Functions and Their Ranges

A composite function is formed when the output of one function becomes the input of another. In \(\mathrm{g}\mathrm{f}\), apply \(\mathrm{f}\) first and then \(\mathrm{g}\): \(x\mapsto\mathrm{f}(x)\mapsto\mathrm{g}(\mathrm{f}(x))\).

In the two-archer analogy, the first archer represents \(\mathrm{f}\) and the second represents \(\mathrm{g}\). Every output passed by the first must be catchable by the second: \(R_{\mathrm{f}}\subseteq D_{\mathrm{g}}\).

The two-archer condition below describes composition on the whole declared domain of the first function. When some outputs miss the second domain, restrict to the inputs that do land in it; the natural-domain composition can still exist.

• For the composition \(\mathrm g\circ\mathrm f\) to be defined on all of \(D_{\mathrm f}\), require \(R_{\mathrm f}\subseteq D_{\mathrm g}\).
• In that whole-domain case, \(D_{\mathrm g\circ\mathrm f}=D_{\mathrm f}\). Otherwise its natural domain is the set of \(x\in D_{\mathrm f}\) for which \(\mathrm f(x)\in D_{\mathrm g}\).
• \(R_{\mathrm{g}\mathrm{f}}\) is not necessarily equal to \(R_{\mathrm{g}}\); equality can occur for particular functions and domains.

Finding the range of a composite function

1. Use \(R_{\mathrm{f}}\) as the effective domain for \(\mathrm{g}\).
2. The outputs attained by \(\mathrm{g}\) on that effective domain form \(R_{\mathrm{g}\mathrm{f}}\).

For the displayed graphs, \(R_{\mathrm{f}}=[a,\infty)\). Applying \(\mathrm{g}\) on that interval gives \(R_{\mathrm{g}\mathrm{f}}=[\mathrm{g}(a),k)\); the lower endpoint is attained and the horizontal asymptote is not.