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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).