Question

Use the guidelines of section 4.4 to sketch the curve

21.\(y = {\sin ^{ - 1}}\left( {\frac{1}{x}} \right)\)

Short answer

The graph of the given function is drawn.

Step-by-step solution

Domain of \(y\)

Consider the given function as \(f\left( x \right) = {\sin ^{ - 1}}\left( {\frac{1}{x}} \right)\). The function is defined at x=1 and

x=-1.

Thus, the domain is \(\left\{ {x:x \le - 1,x \ge 1} \right\}\).

Intercepts of \(y\)

The y-intercept is the value of y at x=0. Similarly, x-intercept is the value of x at y=0.

For the given function there is no intercepts.

Symmetry of \(y\)

Consider the given function as \(f\left( x \right) = 4x - \tan x\).

Substitute x by –x then,\(f\left( { - x} \right) = {\sin ^{ - 1}}\left( { - \frac{1}{x}} \right)\).

Here, \(f\left( { - x} \right)\) is not equal to \( - f\left( x \right)\) or \(f\left( x \right)\) implies f is neither an even function nor an odd function.

Asymptotes of \(y\)

The function has no vertical asymptote and has horizontal asymptote at \(y = 0\).

Local maximum and minimum

The first derivative of the given function is\(f'\left( x \right) = \frac{{ - 1}}{{{x^2}\sqrt {1 - \frac{1}{{{x^2}}}} }}\)

The derivative of f at zero is not defined.

So, the function has no local maxima or minima.

Concavity and inflection point

The second derivative of the given function is\(f''\left( x \right) = \frac{{2{x^2} - 1}}{{{x^2}\left( {{x^2} - 1} \right)\sqrt {{x^2} - 1} }}\).

So, the function is concave downwards at \(\left( { - \infty , - 1} \right)\left( {1,\infty } \right)\).

Graph of \(y\)

Draw the graph of the given function as shown below:

Therefore, the graph of the given function is drawn.