Problem 19 - Precalculus with Limits

Question

In Exercises 19-36, determine whether the equation represents \(y\) as a function of \(x\). \(x^2 + y^2 = 4\)

Step-by-Step Solution

Answer

The equation \(x^2 + y^2 = 4\) does not represent \(y\) as a function of \(x\).

Step 1: Understanding the concept of function

Review the concept of a function. In mathematics, a function is a rule that assigns each input exactly one output. This means that for every \(x\), there should be one and only one corresponding \(y\).

Step 2: Rewrite the given equation for \(y\)

Now let's rewrite the equation, isolating \(y\) on one side of the equation. This will give us \(y^2 = 4 - x^2\). To solve for \(y\), we find its square root, which results in \(y = \sqrt{4 - x^2}\) and \(y = -\sqrt{4 - x^2}\). Note that the square root of any square \(a^2\) results in \(a\) and \(-a\).

Step 3: Determine if \(y\) is a function of \(x\)

Looking at the output when attempting to solve the equation for \(y\), we got \(y = \sqrt{4 - x^2}\) and \(y = -\sqrt{4 - x^2}\), which means for some \(x\) values, there are indeed two possible \(y\) answers. Therefore, the equation \(x^2 + y^2 = 4\) does not represent \(y\) as a function of \(x\), because for given \(x\) values, more than one \(y\) can sometimes be produced.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and formulas.

• These symbols are known as variables. They are often used to represent unknowns or to generalize arithmetic operations.
• In algebraic expressions, you can perform operations such as addition, subtraction, multiplication, and division.

For many students, algebra serves as a foundation for more advanced topics in mathematics, including calculus and geometry. One essential aspect of algebra is solving equations, which we will discuss further. Algebra allows us to express relationships and patterns mathematically, making it integral to problem-solving in math.

Equation Solving

Equation solving in mathematics is the process of finding the value of an unknown variable that makes the equation true. To solve equations efficiently, variables are isolated on one side, and constants on the other. This often involves:

• Applying operations equally to both sides of the equation, such as adding, subtracting, multiplying, or dividing.
• Rewriting the equation, as seen in the example, where we isolated y in the equation \(x^2 + y^2 = 4\) to become \(y^2 = 4 - x^2\).
• Taking the square root as part of "solving for y," considering both the positive and negative roots.

This process highlights why equation solving is considered a vital skill. It allows us to not only find numerical solutions but also understand how different variables interact within the equation.

Function Representation

In mathematics, a function is a representation that shows the relationship between input and output, usually denoted as \(y = f(x)\). A primary characteristic of a function is that for each input (x-value), there is exactly one output (y-value). This ensures consistency and predictability in function representation.

• The exercise given demonstrated that \(x^2 + y^2 = 4\) does not satisfy this rule because isolating \(y\) results in two possible outputs: \(y = \sqrt{4 - x^2}\) and \(y = -\sqrt{4 - x^2}\).
• This multiple "y" for a single "x" outcome disqualifies it from being a function in terms of strict mathematical definition.

Function representation is crucial because it simplifies complex relationships and ensures a linear and predictable pattern of mapping values between sets.Functions can be represented in various forms, such as equations, tables, graphs, or real-world contexts, making them versatile tools for analysis.