Problem 15 - College Algebra Essentials

Question

Determine whether each equation defines \(y\) as a function of \(x .\) $$ x^{2}+y^{2}=16 $$

Step-by-Step Solution

Answer

No, the given equation does not define y as a function of x. For an equation to define \(y\) as a function of \(x\), each \(x\) value must correspond to exactly one \(y\) value. In the case of the given equation, \(x^{2}+y^{2}=16\), each \(x\) value corresponds to two \(y\) values (one positive and one negative), thus, it isn't a function.

Step 1: Standard form of a circle equation

Recognize that the given equation \(x^{2}+y^{2}=16\) is a standard form of a circle equation. The radius \(r\) can be determined by the square root of the constant, in this case 16. Thus, \(r=4\).

Step 2: Solve for y

In order to see if y is a function of x, solve the equation for \(y\). The given equation is \(x^{2}+y^{2}=16\). This can be rearranged to solve for \(y\) by the operation \(y^{2}=16-x^{2}\). Taking the square root of both sides to solve for \(y\) results in \(y = \pm\sqrt{16-x^{2}}\).

Step 3: Determine if y is a function of x

The presence of a \(+\) and \(-\) in the solution indicates that for a given \(x\), there are two possible values of \(y\): one positive and one negative. This means that multiple output values exist for a single input value which defies the definition of a function. Thus, it can be concluded that the given equation does not define \(y\) as a function of \(x\).

Key Concepts

Circle Equation

A circle equation in the standard form is represented as \(x^2 + y^2 = r^2\) where \(r\) is the radius of the circle. In this form, both \(x\) and \(y\) are squared, indicating the symmetry about the center of the circle, which is positioned at the origin \((0,0)\) in this case. This equation represents all the points \((x, y)\) that are exactly \(r\) units away from the center. It’s important because it provides a simple yet complete description of the circle in a Cartesian plane. By examining and manipulating this equation, you can infer many characteristics of the circle, such as radius, position, and even transformations when the equation changes.

Radius Calculation

Calculating the radius from the standard circle equation is straightforward. In an equation form as \(x^2 + y^2 = r^2\), the radius \(r\) can be found by taking the square root of the constant on the other side of the equation. For example, given the equation \(x^2 + y^2 = 16\), you can determine that \(r = \sqrt{16} = 4\).

• The radius is always a positive number, as it represents distance.
• This distance is from the center of the circle, which affects the size of the circle but not its shape.

Remember, knowing the radius is crucial for drawing, analyzing, and utilizing the circle in various mathematical applications.

Solving Equations

Solving equations, especially those that involve squares, involves finding unknown variables while maintaining the equation's balance. For a given circle equation like \(x^2 + y^2 = 16\), solving for \(y\) involves isolating the \(y^2\) term first, resulting in \(y^2 = 16 - x^2\). To further simplify, take the square root of both sides:- This produces \(y = \pm\sqrt{16 - x^2}\), indicating two possible values: positive and negative.When solving, it's also essential to consider all solutions. Both \(+\sqrt{}\) and \(-\sqrt{}\) solutions must be considered, as they represent the upper and lower parts of the circle, respectively.

Mathematical Definitions

In mathematics, understanding definitions is pivotal to grasping broader concepts. For example, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. This definition starkly contrasts with a circle's equation like \(x^2 + y^2 = 16\), which does not fit the criteria for being a function.

• A function must have only one output for each input; however, a circle's equation may give two outputs (positive and negative) for a single input \(x\).
• Understanding these fundamental definitions helps in determining and analyzing various mathematical phenomena.

With these definitions in mind, we can quickly identify whether given equations describe functions or not, such as concluding that the circle equation does not define \(y\) as a function of \(x\).