Reciprocal Functions

Pick a function $f$. Its reciprocal is $\frac{1}{f(x)}$, and it is defined at every $x$ except where $f(x)=0$ — division by zero is undefined, and that restriction must be stated every single time. Everywhere else, three things are true at once: the reciprocal keeps the same sign as $f$ (a positive divided by a positive is positive; $1$ divided by a negative is negative), large outputs of $f$ become outputs of $\frac{1}{f(x)}$ close to zero, and outputs of $f$ close to zero become enormous outputs of $\frac{1}{f(x)}$.

Suppose $f(c)=0$. As $x$ creeps towards $c$, the value $f(x)$ shrinks towards $0$ but never quite arrives, so $\frac{1}{f(x)}$ grows without bound — towards $+\infty$ if $f$ is approaching $0$ from above, towards $-\infty$ if from below. The graph hugs the vertical line $x=c$ forever without touching it. That line is a vertical asymptote, and it is not part of the graph; it is a guide line. At the other end, as $x\to \pm \infty$ the values of a linear or quadratic $f$ grow without bound, so $\frac{1}{f(x)}\to 0$. The reciprocal of any non-constant polynomial therefore also has the horizontal asymptote $y=0$, and it never reaches that one either, because $\frac{1}{f(x)}=0$ has no solution: a fraction with numerator $1$ is never zero.

A point is invariant if it sits on both $y=f(x)$ and $y=\frac{1}{f(x)}$ — flipping the function leaves it exactly where it was. That requires $f(x)=\frac{1}{f(x)}$, so $[f(x){]}^2=1$, so $f(x)=1$ or $f(x)=-1$. Those are the only possibilities. Solving those two equations gives you two free points to anchor your sketch, which is why you should always do it.

Graphing Reciprocal Functions (1 of 4: Linear example)

The same rules apply, but a quadratic can have two, one, or no real zeros, so its reciprocal can have two, one, or no vertical asymptotes. Factor or use the discriminant first. One extra feature appears: wherever $f$ has a non-zero minimum, its reciprocal has a local maximum at the same $x$-value (and vice versa), because the smallest positive denominator gives the largest quotient.

• Two distinct real zeros (e.g. $f(x)=x^2-4$): two vertical asymptotes, at $x=-2$ and $x=2$.
• One repeated zero (e.g. $f(x)={(x-1)}^2$): one vertical asymptote at $x=1$, and the reciprocal goes to $+\infty$ on both sides because $f$ is positive on both sides.
• No real zeros (e.g. $f(x)=x^2+1$): no vertical asymptote at all; the reciprocal is a smooth hump defined for every real $x$.
• When the vertex of $f$ is not on the $x$-axis, it sits directly above or below the turning point of the reciprocal, at the same $x$-value; a vertex on the $x$-axis (a repeated zero) becomes a vertical asymptote instead.

How to Sketch Rational Reciprocal of Quadratic Function