Question

(a) Graph the function \(f\left( x \right) = \sin \left( {{{\sin }^{ - {\bf{1}}}}x} \right)\) and explain the appearance of the graph.

(b) Graph the function \(g\left( x \right) = {\sin ^{ - {\bf{1}}}}\left( {\sin x} \right)\). How do you explain the appearance of this graph?

Short answer

a) The identity function \(y = x\) on the restricted domain is obtained because one function undoes what the other one does.

b) The sine function is monotonic on each interval and, therefore, in \(g\)(but in a linear fashion).

Step-by-step solution

Graph the function \(f\left( x \right) = \sin \left( {{{\sin }^{ - 1}}x} \right)\)

a)

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

1. Open the graphing calculator. Enter the equation\(f\left( x \right) = \sin \left( {{{\sin }^{ - 1}}x} \right)\)in the tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the functions as shown below:

The identity function \(y = x\) on the restricted domain is obtained because one function undoes what the other one does.

Graph the function \(g\left( x \right) = {\sin ^{ - 1}}\left( {\sin x} \right)\)

b)

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

1. Open the graphing calculator. Enter the equation\(g\left( x \right) = {\sin ^{ - 1}}\left( {\sin x} \right)\)in the tab.

2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the functions as shown below:

It is the same as part (a) but with the domain \(\mathbb{R}\). The equation for the function \(g\) on intervals of the form \(\left[ { - \frac{\pi }{2} + \pi n,\frac{\pi }{2} + \pi n} \right]\), for each integer \(n\), can be obtained by using \(g\left( x \right) = {\left( { - 1} \right)^n}x + {\left( { - 1} \right)^{n + 1}}n\pi \). The sine function is monotonic on each of the intervals and, therefore, in \(g\)(but in a linear fashion).