The Reciprocal Function

Definition The reciprocal function is

A point $(x, y)$ lies on its graph exactly when

since for $x \ne 0$ the equation $xy = 1$ is equivalent to $y = \frac{1}{x}$, and no point with $x = 0$ satisfies it.

Definition The domain of $f(x) = \frac{1}{x}$ is the set of all nonzero real numbers,

The input $0$ is excluded because no real $y$ satisfies $0 \cdot y = 1$. Any formula built from $\frac{1}{x}$ keeps this restriction, and a scaled version $\frac{k}{x}$ with $k \ne 0$ has the same domain.

Definition The range of $f(x) = \frac{1}{x}$ is the set of all nonzero real numbers. For every $y \ne 0$ the input $x = \frac{1}{y}$ is nonzero and gives $f\!\left(\frac{1}{y}\right) = y$. The output $0$ never occurs, since $\frac{1}{x} = 0$ would mean $1 = 0 \cdot x = 0$.

Definition A quantity $y$ is inversely proportional to a nonzero quantity $x$ if there is a constant $k \ne 0$ with

The constant $k$ is the constant of proportionality. The reciprocal function is the case $k = 1$.

Definition The graph of $y = \frac{1}{x}$ consists of two separate pieces called branches. For $x > 0$ the output is positive, so that branch lies in the first quadrant. For $x < 0$ the output is negative, so that branch lies in the third quadrant. Since no point has $x = 0$, the branches never connect. On each of the intervals $(-\infty, 0)$ and $(0, \infty)$ the function is decreasing, but it is not decreasing on its whole domain.

Definition A function is odd, and its graph has origin symmetry, if $-x$ is in the domain whenever $x$ is and $f(-x) = -f(x)$ for every $x$ in the domain. The reciprocal function is odd, because for $x \ne 0$

Definition The statement $f(x) \to \infty$ as $x \to a^+$ means that the outputs become and stay as large as you like for all inputs close enough to $a$ on the right. Precisely, for every number $M > 0$ there is a distance $\delta > 0$ such that

For $f(x) \to -\infty$ replace $f(x) > M$ by $f(x) < -M$, and for $x \to a^-$ use the inputs with $a - \delta < x < a$ instead. The line $x = a$ is a vertical asymptote of the graph of $f$ if at least one of these happens, that is, $f(x) \to \infty$ or $f(x) \to -\infty$ as $x \to a^+$ or as $x \to a^-$. For $f(x) = \frac{1}{x}$ the line $x = 0$ is a vertical asymptote, with

Given $M > 0$, the distance $\delta = \frac{1}{M}$ works on both sides, because $0 < x < \frac{1}{M}$ gives $\frac{1}{x} > M$ and $-\frac{1}{M} < x < 0$ gives $\frac{1}{x} < -M$.

Definition The statement $f(x) \to L$ as $x \to \infty$ means that the outputs become and stay as close to $L$ as you like for all large enough inputs. Precisely, for every number $\varepsilon > 0$ there is a number $N$ such that

For $x \to -\infty$ use the inputs with $x < -N$ instead. The line $y = L$ is a horizontal asymptote of the graph of $f$ if $f(x) \to L$ as $x \to \infty$ or as $x \to -\infty$. For $f(x) = \frac{1}{x}$ the line $y = 0$ is a horizontal asymptote in both directions. Given $\varepsilon > 0$, take $N = \frac{1}{\varepsilon}$, because $\lvert x \rvert > \frac{1}{\varepsilon}$ gives $\left\lvert \frac{1}{x} \right\rvert < \varepsilon$. The graph never reaches the line, since $0$ is not in the range.