To sketch the reciprocal trigonometric functions, we could use a table of values approach as we did with primary trigonometric ratios in a previous module.

We also employed a table of values approach when we sketched \(y = x^2\), \(y = \sqrt{x}\), \(y = 2^x\), \(y = \lvert x \rvert\), and \(y = \dfrac{1}{x}\), earlier in this course.

However, for the reciprocal function \(y=\dfrac{1}{x}\), we provided a second approach.

We will review this second approach and we will generalize it in this module to sketch the three reciprocal trigonometric functions.

Sketch the graph of \(f(x) = \dfrac{1}{x}\)

1. Sketch the line \(g(x) = x\). In general, given \(f(x) = \dfrac{1}{g(x)}\), we graph the denominator function \(y = g(x)\).

2. When \(g(x) = \pm 1\), \(f(x) = \dfrac{1}{g(x)} = \pm 1\). In this example, this occurs when \(x = \pm 1\). The points \((1,1)\) and \((-1,-1)\) are on both \(g(x) = x\) and \(f(x) = \dfrac{1}{x}\). Plot these two points.

3. If \(g(x)=0\), \(f(x) = \dfrac{1}{g(x)}\) is undefined. Draw a vertical asymptote along the \(y\)-axis.

4. As values of \(x\) go from \(1\) to \(+ \infty\), \(g(x)\), the denominator\(\phantom{\dfrac{2}{3}}\) function, approaches \(\infty\) and so \(f(x) = \dfrac{1}{g(x)}\) gets closer and closer to zero, the \(x\)-axis, from above.

Sketch the graph of \(f(x) = \dfrac{1}{x}\)

5. As values of \(x\) go from \(-1\) to \(- \infty\), \(g(x)\), the denominator function, approaches \(- \infty\) and so \(f(x) = \dfrac{1}{g(x)}\) gets closer and closer to zero, the \(x\)-axis, from below.

6. Draw in the horizontal asymptote along the \(x\)-axis.

7. As values of \(x\) start from \(-1\) and approach \(0\) from the left, \(g(x)\), the denominator function, approaches \(0\) from below and so \(f(x) = \dfrac{1}{g(x)}\) approaches \(- \infty\).

8. As values of \(x\) start from \(1\) and approach \(0\) from the right, \(g(x)\), the denominator function, approaches \(0\) from above and so \(f(x) = \dfrac{1}{g(x)}\) approaches \(\infty\).

General steps for sketching \(f(x) = \dfrac{1}{g(x)}\)

1. Sketch the function \(y = g(x)\).

2. Identify the values of \(x\) where \(g(x) = 1\) or \(g(x) = -1\). At these points \(f(x) = g(x)\). That is, these points are on both \(f(x)\) and \(g(x)\). These points are called fixed points or static points.

3. Identify the \(x\)-intercepts of \(g(x)\). At these points, \(f(x)\) is undefined. There will be vertical asymptotes for these values of \(x\).

4. If required, determine what happens to the reciprocal function as \(x\) approaches the vertical asymptotes from the left and from the right.

5. If required, determine the end behaviour of \(f(x)\).