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.

Short answer

The graph is a function.

Part (a) The domain of the function is the set of all real numbers and the range of the function is $\left\{y\overline{)y>0}\right\}$.

Part (b) The intercept is$\left(0,1\right)$.

Part (c) There is no symmetry.

Step-by-step solution

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 the set of all real numbers.

The range of the function is$\left\{y\overline{)y>0}\right\}$.

Part (b) Step 1.

$\left(0,1\right)$is the only intercept on the graph.

Part (c) Step 1.

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 and also there is no projection along a line through the origin so that the distances from the origin are equal.