Question

Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\operatorname{arcsec} 2 x $$

Short answer

The function \(f(x) = \operatorname{arcsec}(2x)\) is a horizontally compressed version of the arcsecant function. The domain of the function is \(x \leq -1\) or \(x \geq 1\), and the range is \(0 \leq y < \frac{\pi}{2}\) or \(\frac{\pi}{2} < y \leq \pi\), excluding \(y = \frac{\pi}{2}\). The graph includes vertical asymptotes at \(x = -\frac{1}{2}\), \(x = -\frac{1}{4}\), \(x = \frac{1}{4}\), \(x = \frac{1}{2}\).

Step-by-step solution

Understand the secant function

The secant function, \(sec(x)\), has an undefined value at \(x = (2n + 1) \frac{\pi}{2}\), where \(n\) is an integer. This is because at these \(x\) values, the cosine function is 0 and the secant function is the reciprocal of the cosine function. This means that the secant function will have vertical asymptotes at these \(x\) values. The secant function is positive in the first and fourth quadrants where the cosine function is also positive, and negative in the second and third quadrants where the cosine function is negative. The period of the secant function is \(2\pi\) and it increases from negative infinity to 1, jumps to negative 1 and increases again to positive infinity within each period. The secant wave has an amplitude of 1.

Inverse the secant function

The arcsecant function, denoted as \(arcsec(x)\) or \(sec^{-1}(x)\), is the inverse of the secant function. This means that it reflects the secant function along the line \(y = x\). Due to the asymptotes of the secant function, the domain of the arcsecant function is \(x \leq -1\) or \(x \geq 1\). The range of the arcsecant function is \(0 \leq y < \frac{\pi}{2}\) or \(\frac{\pi}{2} < y \leq \pi\), excluding \(y = \frac{\pi}{2}\), as this is where the original secant function has vertical asymptotes. Just like the secant function, the arcsecant function is positive in the first and fourth quadrants, and negative in the second and third quadrants.

Apply the function transformation

The function given in the problem is \(f(x) = \operatorname{arcsec}(2x)\). The \(2x\) inside the function means the input to the function is being scaled by a factor of 2. This means the graph will be horizontally compressed by a factor of 1/2. Therefore, the period of the function will be halved, changing from \(2\pi\) to \(\pi\).

Sketch the graph

Start by drawing the asymptotes of the function (\(x = -\frac{1}{2}\), \(x = -\frac{1}{4}\), \(x = \frac{1}{4}\), \(x = \frac{1}{2}\)), there are some points located on the \(x\) axes that the arcsecant function will not pass including 0. The graph passes the points (-1, -pi), (-0.5, 0), and (1, pi). You can then sketch a smooth curve that goes through these points and approaches the asymptotes.

Verify the graph using a graphing utility

Upon completing the sketch, use a graphing utility to verify your graph. Input the function into the graphing utility, ensure to adjust the window settings to accurately view the graph. Compare your sketch with the graph provided by the utility tool.