Goal: Sketch the graph of \(f(x)=\dfrac{1}{x}\). State the domain and range.

The function takes a value of \(x\) and determines its reciprocal.

Only one value of \(x\), that is \(x=0\), would not be allowed since division by \(0\) is undefined.

We make a table of values as before.

Plot the points and then join them with a smooth curve.

The final appearance of the graph is not entirely obvious.

It is not obvious what happens for values of \(x\), other than \(0\), between \(-1\) and \(1\).

Similarly, it is not obvious what happens to the function when \(x\) approaches positive or negative infinity.

As \(x\) gets closer and closer to \(0\) from the left side, the reciprocal approaches negative infinity.

We write this using the notation, “\(\text {As }x\rightarrow 0^-,\ \dfrac{1}{x}\rightarrow -\infty\).”

Similarly, as \(x\) gets closer and closer to \(0\) from the right side, the reciprocal approaches positive infinity.

We write this using the notation, “\(\text{As }x\rightarrow 0^+,\ \dfrac{1}{x}\rightarrow +\infty\).”

Some values are shown in the following tables of values.

The curve gets closer and closer to the \(y\)-axis but never crosses it.

As \(x\) approaches negative infinity, the reciprocal gets closer to \(0\) and is negative.

We write this using the notation, “As \(x\rightarrow -\infty,\ \dfrac{1}{x}\rightarrow 0\) from below.”

Similarly, as \(x\) approaches positive infinity, the reciprocal gets closer and closer to \(0\) but stays positive.

We write this using the notation, “As \(x\rightarrow +\infty,\ \dfrac{1}{x}\rightarrow 0\) from above.”

Some values are shown in the following tables of values.

The curve gets closer and closer to the \(x\)-axis but never crosses it.