Problem 74 - Algebra and Trigonometry

Question

a. Graph the restricted cotangent function, \(y=\cot x,\) by restricting \(x\) to the interval \((0, \pi)\). b. Use the horizontal line test to explain why the restricted cotangent function has an inverse function. c. Use the graph of the restricted cotangent function to graph \(y=\cot ^{-1} x\).

Step-by-Step Solution

Answer

To answer the given problem: a. The restricted cotangent graph is an open curve from positive to negative infinity with asymptotes at 0 and \(π\), and crossing the x-axis at \(π/2\). b. The horizontal line test shows that each horizontal line only intersects the curve once, so there is an inverse function. c. The inverse cotangent graph is a reflection of the original curve along the line \(y = x\), with vertical asymptotes now at \(y = 0\) and \(y = π\), and crossing the y-axis at \(y = π/2\).

Step 1: Graphing the Restricted Cotangent Function

First, let's draw the graph of the restricted cotangent function \(y = cot(x)\) for \(x \) in \((0, π)\). Start by identifying key points, like the points of discontinuity at \(x = 0\) and \(x = π\), and the x-intercept at \(x = π/2\). We know the function approaches positive and negative infinity at the endpoints, and it crosses the x-axis at \(π/2\). Connect these points to draw the graph correctly.

Step 2: Applying the Horizontal Line Test

Now, apply the horizontal line test to the graph from Step 1. This test checks whether any horizontal line crosses the graph more than once. If it does, the function does not have an inverse - its not bijective. For \(y = cot(x)\) restricted to \((0, π)\), each horizontal line intersects the curve at most one time, indicating that the function does indeed have an inverse.

Step 3: Graphing the Inverse Cotangent Function

Finally, you can graph the inverse cotangent function, \(y = cot^{-1}(x)\), using the graph from step 1. The graph of an inverse function is obtained by reflecting the graph of the original function across the line \(y = x\). This changes all of the (x, y) points on the original graph to (y, x) points on the new graph. That is, horizontal asymptotes become vertical asymptotes, and the curve changes its orientation. Therefore, the inverse cotangent function will have vertical asymptotes at \(y = 0\) and \(y = π\), and it will cross the y-axis at \(y = π/2\).