Question

In Exercises 115 and \(116,\) find the point of intersection of the graphs of the functions. $$ \begin{array}{l} y=\arccos x \\ y=\arctan x \end{array} $$

Short answer

The intersection point of the graphs of the functions \(y = \arccos x\) and \(y = \arctan x\) is at (0, \(\pi/2\)).

Step-by-step solution

Equate the two functions

To find the intersection point of these two graphs, we must find the value of \(x\) where the two functions are equal. In other words, we solve the equation \(\arccos x = \arctan x\).

Solve for \(x\)

When analyzing the functions, we know that \(\arccos x\) ranges from 0 to \(\pi\), and \(\arctan x\) ranges from \(-\pi/2\) to \(\pi /2\). Looking at the overlapping interval, it's from 0 to \(\pi /2\). We also know that \(\arccos x\) is decreasing from \(\pi\) to 0 on the interval [-1,1], and \(\arctan x\) is increasing from -\(\pi /2\) to \(\pi /2\) on the same interval. Therefore, these two functions will intersect once, at \(x=0\). This can be further verified by making a graph of these two functions.

Find \(y\)-coordinates

To find the intersection point, we know the \(x\)-coordinate is 0. We need to find the \(y\)-coordinates by substituting \(x=0\) into each of the original functions. When we substitute \(x=0\) into \(y = \arccos x\) we get \(y = \pi /2\), and when we substitute \(x=0\) into \(y = \arctan x\) we get \(y = 0\). Since we are finding a common point for both functions, we take the \(\pi /2\) value.