Question

a. determine whether the equation represents a function.
$(x + 2)^2 + (y - 3)^2 = 9$
a. yes - it is a function
b. no - it is not a function

Explanation:

Step1: Recall the definition of a function

A function is a relation where each input (x - value) has exactly one output (y - value). We can use the vertical line test: if any vertical line intersects the graph of the relation more than once, it is not a function.

Step2: Analyze the given equation

The equation \((x + 2)^{2}+(y - 3)^{2}=9\) is the standard form of a circle with center \((- 2,3)\) and radius \(r = 3\) (since the standard form of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius).

For a circle, if we draw a vertical line (parallel to the y - axis) through the circle, the vertical line will intersect the circle at two points (except for the vertical lines that are tangent to the circle, but in general, most vertical lines through the circle will intersect it at two points).

Mathematically, if we try to solve for \(y\) in terms of \(x\):
\[

$$\begin{align*} (x + 2)^{2}+(y - 3)^{2}&=9\\ (y - 3)^{2}&=9-(x + 2)^{2}\\ y-3&=\pm\sqrt{9-(x + 2)^{2}}\\ y&=3\pm\sqrt{9-(x + 2)^{2}} \end{align*}$$

\]
For a given \(x\) (such that \(9-(x + 2)^{2}\geq0\), i.e., \(-5\leq x\leq1\)), there are two values of \(y\) (one with the plus sign and one with the minus sign). This means that there are \(x\) - values that correspond to two different \(y\) - values, so the relation does not satisfy the definition of a function.

Answer:

B. No - it is not a function