Q. 21 - 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 all real numbers and the range of the function is $\left\{y\overline{)y\ge -3}\right\}$.

Part (b) The intercepts are $\left(0, 9\right), \left(1, 0\right) and \left(3, 0\right)$.

Part (c) There is no symmetry.

Part (a) Step 1. Given Information.

Consider the given graph:

Part (a) Step 1. Find domain and range.

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 all real numbers.

The range of the function is $\left\{y\overline{)y\ge -3}\right\}$.

Part (b) Step 1.

The intercepts on the graph are$\left(0, 9\right), \left(1, 0\right) and \left(3, 0\right)$.

Part (c) Step 1. Find the symmetry.

Vertical-line Test: A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph at most one point.

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 (c) Step 2. Explanation.

The graph is not symmetric with respect to the x-axis as there is no part of the graph above the x-axis which is a reflection or mirror image of the part below it, and vice versa.

The graph is not symmetric with respect to the y-axis as there is no part of the graph to the right of the y-axis which is a reflection of the part to the left of it, and vice versa.

There is no symmetry with respect to the origin as there is no reflection about the y-axis which is followed by a reflection about the x-axis also there is no projection along a line through the origin so that the distances from the origin are equal.