Problem 61 - Algebra and Trigonometry

Question

Determine whether the equation defines y as a function of x. (See Example 9.) \(x+y^{2}=9\)

Step-by-Step Solution

Answer

No, \(y\) is not a function of \(x\).

Step 1: Isolate y

To determine if \(y\) can be expressed as a function of \(x\), we first try to isolate \(y\) in the equation. Start by moving \(x\) to the right side of the equation: \(y^2 = 9 - x\).

Step 2: Solve for y

Take the square root of both sides to solve for \(y\). This gives \(y = \sqrt{9 - x}\) and \(y = -\sqrt{9 - x}\).

Step 3: Determine if y is a function

A function assigns exactly one output to each input. Here, for each \(x\) between \(-\infty\) and \(9\), there are two possible values of \(y\) (one positive and one negative). Thus, \(y\) is not a function of \(x\) because each \(x\) does not correspond to exactly one \(y\).

Key Concepts

Understanding Function Definition

In mathematics, a function is a special type of relation. It uniquely associates elements of one set (the domain) with exactly one element of another set (the range). In simple terms, a function takes an input and produces a single output.
Functions are often written in the form \(y = f(x)\), where \(x\) is the input variable and \(y\) is the output. The key characteristic of a function is that each input corresponds to exactly one output.

• For example, consider the function \(f(x) = x^2\). For each \(x\), there is exactly one \(y\) value, \(x^2\).
• If you input \(2\), the output is \(4\), and if you input \(3\), the output is \(9\).

In the problem above, we must determine if \(y\) is a function of \(x\). If at any point a single \(x\) value gives multiple \(y\) values, it is not a function.

Exploring Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. They can involve numbers, variables, and operators such as addition, subtraction, and more.
The goal is to find the value of unknown variables that satisfy the equation. In the exercise, we start with the equation \(x+y^{2}=9\). The objective is to manipulate this equation to find \(y\) in terms of \(x\).

• Isolating \(y\): Move terms involving \(x\) to the other side, resulting in \(y^2 = 9 - x\).
• This transformation is crucial because it simplifies the algebraic equation to directly examine if \(y\) can behave as a function of \(x\).

Understanding how to rearrange and manipulate algebraic equations is fundamental in determining relationships between variables, such as whether a relation defines a function.

The Intricacies of Square Roots

The square root is a mathematical function that, when applied to a number, yields a value which, when multiplied by itself, gives the original number. The square root operation is denoted by the radical symbol \(\sqrt{}\).
When solving equations involving square roots, it's critical to remember that each positive real number has two square roots—one positive and one negative.

• Example: The square root of 9 is 3, and also -3, since both \(3^2\) and \((-3)^2\) equal 9.
• In the context of our exercise, when solving \(y^2 = 9 - x\), applying the square root gives two results: \(y = \sqrt{9 - x}\) and \(y = -\sqrt{9 - x}\).

This dual result is why \(y\) cannot be considered a function of \(x\) in this instance, as each \(x\) value results in two possible \(y\) outputs.