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Reciprocal functions

A reciprocal function takes the form $$$ y = \frac{\Tblue{a}}{x}$$$ for a number $$\Tblue{a}$$. We can also write the function as $$\Tblue{a}/x$$ or $$\Tblue{a} x^{-1}$$.

$$\displaystyle\frac{\Tblue{1}}{x}$$ and $$\displaystyle\frac{\Tblue{-2}}{x}$$ are reciprocal functions.

The number $$\Tblue{a}$$ is the value of the function at $$x=1$$.

The reciprocal function is defined for $$x\ne0$$. It has no value at $$0$$.

For positive $$x$$, $$y = \Tblue{1}/x$$ becomes bigger as $$x$$ gets smaller. $$$f(1) = 1,\; f(0.5) = 2,\; f(0.1) = 10,\; f(0.01) = 100.$$$ As positive values of $$x$$ tend to $$0$$, the graph gets closer to the upper $$y$$-axis but never crosses it. Similarly, for negative $$x$$, the graph gets closer to the lower $$y$$-axis.

We say that the $$y$$-axis is a vertical asymptote of $$1/x$$.

Graphs of the reciprocal functions $$y= 1/x$$ (left), $$y = 3/x$$ (centre) and $$y = - 2/x$$ (right).
Graphs of the reciprocal functions $$y= 1/x$$ (left), $$y = 3/x$$ (centre) and $$y = - 2/x$$ (right).