Problem 15 - College Algebra

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 \(x\), because for every value of \(x\), there are two corresponding values of \(y\), which is contrary to the definition of a function.

Step 1: Interpret the Given Equation

We begin with the given equation \(x^2 + y^2 = 16\). This equation represents a circle with radius 4 centered at the origin in a two-dimensional Cartesian coordinate system.

Step 2: Isolate y

Subtract \(x^2\) from both sides of the equation. This gives us \(y^2 = 16 - x^2\).

Step 3: Solve for y

To further isolate \(y\), now take the square root of both sides. Remember that the square root of a number can be both positive and negative. This gives us \(y = ±\sqrt{16 - x^2}\).

Step 4: Look for the Definition of a Function

The ± symbol indicates two potential functions: \(y = \sqrt{16 - x^2}\) and \(y = -\sqrt{16 - x^2}\). This means for every value of \(x\), there are two corresponding values of \(y\), which violates the definition of a function that each element in the domain (x-values) corresponds to exactly one element in the range (y-values).

Key Concepts

Circle Equation

A circle equation is a mathematical representation of a circle in a coordinate plane. The standard form of a circle's equation is \[ \left( x - h \right)^2 + \left( y - k \right)^2 = r^2 \] where

• \( (h, k) \) is the center of the circle
• \( r \) is the radius

In this form, the equation reveals the point around which all points on the circle are equidistant. For our specific equation, \( x^2 + y^2 = 16 \), the circle is centered at the origin \( (0, 0) \) with a radius of 4, since \[ r^2 = 16 \rightarrow r = \sqrt{16} = 4. \] By understanding the structure of a circle's equation, it becomes easier to interpret the spatial relationships it defines in a given coordinate system.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane used to graphically represent equations and geometric shapes. It comprises two perpendicular axes which intersect at the origin. These axes divide the plane into four quadrants. The horizontal axis is the x-axis and the vertical axis is the y-axis.

• Points on the plane are identified by pairs of coordinates \((x, y)\).
• The origin is the point \((0, 0)\).
• Positive x-values extend to the right of the origin, and negative ones to the left.
• Positive y-values extend upwards from the origin, and negative ones downwards.

This system makes it possible to conveniently represent algebraic equations and solve them visually. When considering the circle equation \(x^2 + y^2 = 16\), the Cartesian coordinate system helps illustrate that all points \((x, y)\) satisfying the equation are positioned equidistant from the origin, forming a circle.

Definition of a Function

The definition of a function in mathematics is a relation in which each element in the domain corresponds to exactly one element in the range. In more simple terms, a function means that for every input, there is only one output. For the given equation \(x^2 + y^2 = 16\), when isolating \(y\) as \[ y = \pm \sqrt{16 - x^2}, \]this shows that for a single x-value, there can be two possible y-values, one positive and one negative. Thus, it does not meet the criteria to be a function because multiple outputs (y-values) can be derived from a single input (x-value). This understanding is crucial in determining whether equations define functions.

Solving Equations

Solving equations involves finding the values that satisfy the given equation. This usually requires manipulation of the equation to isolate the variable of interest. Let's take the example of the circle equation \(x^2 + y^2 = 16\). To solve for \(y\), follow these steps:

• Subtract \(x^2\) from both sides: \(y^2 = 16 - x^2\).
• Take the square root of both sides: \(y = \pm \sqrt{16 - x^2}\).

The \(\pm\) indicates that two values of \(y\) satisfy the equation for each \(x\) within the interval \([-4, 4]\). While solving, always ensure that the solution meets the original equation's conditions. This is especially important when determining if the equation represents a function, as shown where multiple solutions indicate it does not.