Question

Locate the discontinuities of the function and illustrate by graphing.

39. \(f\left( x \right) = \frac{1}{{\sqrt {1 - \sin x} }}\)

Short answer

The graph of the function illustrates the discontinuities for \(n = - 1\), 0 and 1.

Step-by-step solution

Theorems of continuity

When the functions say\(f\)and\(g\)arecontinuousat some number. Then, the following functions are also continuous:

1. \(f + g\)
2. \(f - g\)
3. \(cf\)
4. \(fg\)
5. \(\frac{f}{g},\,\,{\mathop{\rm if}\nolimits} \,\,g\left( a \right) \ne 0\)

Eachpolynomialiscontinuous on\(\mathbb{R} = \left( { - \infty ,\infty } \right)\). Anyrational functionis continuous in its domain.

When\(f\)and\(g\)are continuous at some number, then thecomposite function\(f \circ g\)is \(\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)\) also continuous.

Use a graphing calculator to graph the function

The procedure to draw the graph of the function by using the graphing calculator is as follows:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(1/{\left( {1 - \sin X} \right)^{1/2}}\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the function \(f\left( x \right) = \frac{1}{{\sqrt {1 - \sin x} }}\) as shown below:

Locate the discontinuities of the function

There is a discontinuity in the function\(f\left( x \right) = \frac{1}{{\sqrt {1 - \sin x} }}\)wherever;

\(\begin{array}{c}1 - \sin x = 0\\\sin x = 1\\x = \frac{\pi }{2} + 2n\pi \end{array}\)

With\(n\)is any integer.

It is observed that the graph illustrates the discontinuities for \(n = - 1\), 0 and 1.