Reciprocal and squared transformations

For the reciprocal graph, a continuous zero approached through nonzero values gives an infinite branch. A vertical asymptote of \(\mathrm{f}\) can give a limiting height \(0\), but its excluded input remains excluded: it does not automatically become an intercept.

Squaring moves heights with \(0<|v|<1\) nearer the axis and heights with \(|v|>1\) farther away; negative heights become positive.

Example

For \(\mathrm{f}(x)=x-2\), the transforms are \(\dfrac1{x-2}\), with asymptotes \(x=2\), \(y=0\), and \((x-2)^2\), with vertex \((2,0)\).

Reading a squared graph

For \(y=[\mathrm f(x)]^2\), retain every permitted input and square its height. Zeros remain zeros and negative branches are reflected upward before their heights are squared. Points at heights \(0\) and \(1\) are fixed; height \(-1\) becomes \(1\).

The square of a range must be found from the whole interval. If \(R_{\mathrm f}=[-2,3]\), then \(R_{[\mathrm f]^2}=[0,9]\), since the original range includes \(0\). Squaring only the two endpoints would incorrectly give \([4,9]\).

If \(\mathrm f(x)\to a\) at an end of the domain, then \([\mathrm f(x)]^2\to a^2\). Both positive and negative infinite branches become positive infinite branches. An original missing input remains missing even when the squared expression simplifies.

Distinguish \([\mathrm f(x)]^2\) from \((\mathrm f\circ\mathrm f)(x)\). For \(\mathrm f(x)=x+1\), these are \((x+1)^2\) and \(x+2\), respectively. Use the given notation and domain to decide which operation is intended.