Reciprocal Graphs

For the reciprocal graph \(y=1/\mathrm{f}(x)\), a point \((x,y)\) with \(y\ne0\) maps to \(\left(x,1/y\right)\).

The domain of the reciprocal is \(D_{\mathrm f}\setminus\{x:\mathrm f(x)=0\}\). Keep every exclusion from the original function.

If \(\mathrm f\) is continuous near an isolated zero \(x=a\), with nonzero values arbitrarily close to \(a\), then \(1/\mathrm f(x)\) becomes unbounded in magnitude as \(x\to a\), giving a vertical asymptote. Where \(\mathrm f(x)\to\pm\infty\), its reciprocal tends to \(0\); an originally excluded input remains excluded and does not become an intercept.

A non-axis minimum becomes a maximum when \(f\) keeps one nonzero sign near the point; continuity there is sufficient.

A non-axis maximum becomes a minimum under the same local sign condition.

A non-zero horizontal asymptote \(y=a\) becomes \(y=1/a\).

Sign and monotonicity

• If \(\mathrm{f}(x)>0\), then \(1/\mathrm{f}(x)>0\); if \(\mathrm{f}(x)<0\), then \(1/\mathrm{f}(x)<0\).
• If \(\mathrm f(x)\to0\) through nonzero values, then \(|1/\mathrm f(x)|\to\infty\). Use the sign on each side to decide whether the reciprocal tends to \(+\infty\) or \(-\infty\).
• If \(\mathrm{f}(x)\) tends to infinity, then \(1/\mathrm{f}(x)\to0\).
• On an interval where \(\mathrm f\) has a constant nonzero sign, increasing portions become decreasing portions under reciprocation, and vice versa.