Question

Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arccos \frac{x}{4} $$

Short answer

The graph of \(f(x) = \arccos \(\frac{x}{4}\)\) starts from point (-4, π), reaches a peak at (0, π/2), and descends to the point (4, 0). This makes the plot decrease across its domain from -4 to 4.

Step-by-step solution

Determine the Domain and Range

First, the domain and range of the given function can be determined. For the arccosine function \(f(x) = \arccos x\), the domain is restricted between -1 and 1 inclusive (i.e. \(x \in \[-1, 1\]\)), and the range falls between 0 and π inclusive (i.e. \(y \in \[0, π\]\)). But for the function \(f(x) = \arccos \(\frac{x}{4}\)\), the domain will fall between -4 and 4 inclusive because the input is now \(x/4\). The range remains the same as the original arccosine function and falls between 0 and π inclusive.

Plot Key Points

Briefly plot key points. Start from the endpoints of the domain, which are -4 and 4. When \(x = -4\), \(f(x) = \arccos(-1) = π\). When \(x = 4\), \(f(x) = \arccos(1) = 0\). Another key point to note is when x = 0, then \(f(x) = \arccos(0) = π/2\). So plot these three points: (-4, π), (0, π/2), and (4, 0).

Sketch the Graph

Finally, sketch the graph of the function. The graph starts from the point (-4, π), reaches a peak at (0, π/2) and then descends to the point (4, 0). It's important to note that the graph of the arccosine function is decreasing on its domain.

Verification with a Graphing Utility

Now input the function \(f(x) = \arccos \(\frac{x}{4}\)\) into a graphing utility to verify whether the sketch matches the curve produced by the graphing utility.