Combining Functions
Building blocks

Composing gets me nowhere
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Warm-up

If \(f(x)=\dfrac{1}{x}\), for \(x\ne 0\), what is \(ff(x)\)?

By \(ff(x)\) we mean the composition of \(f\) with itself, sometimes written as \(f^2(x)\).

\[ff(x)=f\left(\frac{1}{x}\right)=\frac{1}{\left(\frac{1}{x}\right)}=x\]

The composition of \(f\) with itself is a function whose output is the same as the input value. We call \(f\) a self-inverse function. Can you explain why we use this name?

Note that the composition \(ff(x)\) is undefined at \(x=0\) even though the function \(x\) is defined.

Can you find any other functions that behave in this way when composed with themselves?

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Last updated 27-Jan-16
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