Question

The function \(f(x)=\sec^{-1}x\).

(a) List the domain and range.

(b) Sketch a labeled graph.

(c) Discuss the domains and ranges in the context of the unit circle.

Short answer

(a)

Domain: \((-\infty,-1]\cup[1,\infty)\)

Range: \(\left[0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right]\)

(b) The graph is shown.

(c) The domain and range are explained in the steps.

Step-by-step solution

Part (a) Step 1: Given Information

The given function is \(f(x)=\sec^{-1}x\).

Part (a) Step 2: Calculation

The trigonometric functions' domain and range are well known. All we have to do is adjust them in accordance with the function's argument.

Domain: \((-\infty,-1]\cup[1,\infty)\)

Range: \(\left[0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right]\)

Part (b) Step 1: The graph of the function \(f(x)=\sec^{-1}x\).

The graph is,

Notice the graph, the domain, and the range are verified.

Part (c) Step 1: The context of the domain and the range of the unit circle.

The first and second quadrants of the unit circle, with the exception of the portion inside the circle, are the domain and range of this function in the context of the unit circle \(\frac{\pi}{2}\) respectively.