Problem 18 - College Algebra

Question

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). $$ x^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

No, the relation is not a function.

Step 1: Understand the Definition of a Function

A relation is a function if for every input value (usually represented by \(x\)), there is exactly one output value (usually represented by \(y\)). This means, a function can't map a single \(x\)-value to multiple \(y\)-values.

Step 2: Analyze the Given Equation

We are given the equation \(x^2 + y^2 = 9\). This equation resembles the equation of a circle with a radius of 3. A circle can have multiple \(y\)-values (two in most cases) for a given \(x\)-value unless it is a point exactly at the top or bottom of the circle.

Step 3: Check for Multiple Outputs

Rearrange the equation to express \(y\) in terms of \(x\):\[y^2 = 9 - x^2\]Taking the square root of both sides gives:\[y = \pm \sqrt{9 - x^2}\]This shows for each \(x\) (within the domain \(-3 \le x \le 3\)), there are two possible values of \(y\), except when \(x = 3\) or \(x = -3\), which corresponds to the top and bottom points of the circle, respectively.

Step 4: Conclude based on the Analysis

Since there are multiple possible \(y\) values for most \(x\) values within the domain (except the maximum and minimum points), the relation given by \(x^2 + y^2 = 9\) does not define \(y\) as a function of \(x\).

Key Concepts

Circle Equation

In mathematics, a circle equation typically looks like \(x^2 + y^2 = r^2\), where \(r\) represents the radius of the circle. For the equation \(x^2 + y^2 = 9\), the radius \(r\) is 3, since \(3^2 = 9\). This equation represents a circle centered at the origin (0, 0) on the coordinate plane.

• An equation of a circle in this form always describes all the points that are at a distance of \(r\) from the center.
• A circle equation is a perfect example of a geometric layout, where every point on the curve is equidistant from the center.

Understanding circle equations helps recognize the shape and boundary defined by those particular constants in a coordinate plane.

Relation Analysis

Analyzing a relation involves checking how the values of \(x\) relate to values of \(y\). For equations like \(x^2 + y^2 = 9\), when you input a value for \(x\), there can be more than one possible value for \(y\).

• To better analyze, rearrange the equation: \(y^2 = 9 - x^2\).
• Taking the square root means \(y\) can equal \(\pm \sqrt{9 - x^2}\). This means two potential outputs for \(y\) for each \(x\), indicating multiple horizontal slices of the circle.

This understanding of possible outputs is crucial in determining whether the relation is a function or not.

Function Definition

The definition of a function is straightforward: a function is a relation where each input corresponds to exactly one output. It’s like a rule where no two different outputs can arise from one singular input value.

• In mathematical notation, if \(f(x) = y\), for each \(x\) there should only be one \(y\).
• For example, the circle’s equation \(x^2 + y^2 = 9\) does not satisfy this condition since an input \(x\) might return two different \(y\) values.

Thus, while circles are interesting shapes in geometry, their equations generally do not represent functions in terms of standard function definition.

Multiple Outputs

In the context of the circle equation, multiple outputs refer to the fact that for a given \(x\), there can be two corresponding \(y\) values.

• Taking \(x^2 + y^2 = 9\) and solving for \(y\), yields \(y = \pm \sqrt{9 - x^2}\).
• This means for each \(x\) (except at boundaries where \(x = \pm 3\), you’ll find two possible \(y\) values, understood geometrically as the top and bottom parts of the circle.

There is a break in the single output rule here, hence understanding multiple outputs is key when dealing with relations like circles.