Question

consider the following.

graph of a vertical ellipse centered at the origin with y-intercepts at (0, 4) and (0, -4), and x-intercepts at (-1, 0) and (1, 0) (approximate grid lines shown)

use the vertical line test to determine whether the curve is the graph of a function of x.
∘ yes, the curve is a function of x.
∘ no, the curve is not a function of x.

if the curve is a function, state the domain and range. (enter your answers using interval notation. if the curve is not a function enter
domain
range

Explanation:

Step1: Recall Vertical Line Test

The Vertical Line Test states that a graph represents a function of \( x \) if and only if no vertical line intersects the graph more than once.

Step2: Apply the Test to the Graph

Looking at the given graph (an ellipse centered on the y - axis), if we draw a vertical line (for example, the line \( x = 0 \) or any vertical line within the domain of the ellipse), it will intersect the ellipse at two points. This means that there are values of \( x \) (in the domain of the ellipse) for which there are two corresponding \( y \) - values. So, the graph fails the Vertical Line Test.

Answer:

No, the curve is not a function of \( x \).