Porisma

Can one function feed another?

Follow the order of a composite, check every intermediate output, and map the final range.

By the end, you can

  • Read composition from right to left
  • Check whether the first range fits inside the second domain
  • Find the composite domain and range

about 20 minutes for the lesson · practise at your own pace.

The central idea

In gf(x)=g(f(x))gf(x)=g(f(x)), apply ff first, then pass its output to gg. This works on the entire original domain of ff only when Rf⊆DgR_f\subseteq D_g. If even one output is outside the next function’s domain, the composite does not exist on that whole domain. A restriction can define a different composite, but must be stated explicitly.

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Worked example

For f(x)=1−x2f(x)=1-x^2 on [−1,1][-1,1], Rf=[0,1]R_f=[0,1]. With g(t)=t+2g(t)=\sqrt{t}+2 on [0,∞)[0,\infty), every intermediate value is allowed. Thus gf(x)=1−x2+2gf(x)=\sqrt{1-x^2}+2, with domain [−1,1][-1,1] and range [2,3][2,3]. In the opposite order, Rg=[2,∞)R_g=[2,\infty) does not fit inside Df=[−1,1]D_f=[-1,1], so fgfg does not exist on the given domain of gg.

Substitution gives a candidate formula; it does not establish that the composite exists on the printed domain.

Check your understanding

Let a(x)=x2a(x)=x^2 on R\mathbb R and b(t)=t−1b(t)=\sqrt{t-1} on [1,∞)[1,\infty). Decide whether baba and abab exist on their stated domains. Justify each decision using the entire inner range, and give the rule, domain and range of any composite that exists.

Practice questions

9 examples and practice questions, in 2 sections.

More practice

Use the range/domain check on these questions, then work through their existing interactive solutions.

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