Q. 13 - Precalculus Enhanced with Graphing Utilities

Question

Determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin.

Step-by-Step Solution

Answer

The graph is a function.

Part (a) The domain of the function is $\left\{x\overline{)-\mathrm{\pi }\le \mathrm{x}\le \mathrm{\pi }}\right\}$ and the range of the function is $\left\{y\overline{)-1\le y\le 1}\right\}$.

Part (b) The intercepts are $\left(-\frac{\mathrm{\pi }}{2}, 0\right), \left(0, 1\right), and \left(\frac{\mathrm{\pi }}{2}, 0\right)$.

Part (c) The graph is symmetric with respect to y-axis.

Step 1. Given information

The given graph is

Need to determine whether the graph is that of a function by using the vertical-line test.

The graph in the above figure is a function because there is a vertical line that intersects the graph at only one point.

Part (a) Step 1.

When the graph of a function is given, its domain may be viewed as the shadow created by the graph on the x-axis by vertical beams of light. Its range can be viewed as the shadow created by the graph on the y-axis by horizontal beams of light.

The domain of the function is $\left\{x\overline{)-\mathrm{\pi }\le \mathrm{x}\le \mathrm{\pi }}\right\}$.

The range of the function is $\left\{y\overline{)-1\le y\le 1}\right\}$.

Part (b) Step 1.

The intercepts are $\left(-\frac{\mathrm{\pi }}{2}, 0\right), \left(0, 1\right), and \left(\frac{\mathrm{\pi }}{2}, 0\right)$.

Part (c) Step 1.

The graph is symmetric with respect to the y-axis as the part of the graph to the right of the y-axis is a reflection of the part to the left of it, and vice versa.