Problem 13 - College Algebra

Question

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

Step-by-Step Solution

Answer

Yes, the equation \(x^{2}+y-16=0\) defines y as a function of x.

Step 1: Write the given equation

The given equation is \(x^{2}+y-16=0\).

Step 2: Solve the equation for y

By rearranging the terms of our equation, we will isolate y on one side of the equation. So, \(y=16-x^{2}\)

Step 3: Examine the obtained equation for y

After rearranging, we got \(y=16-x^{2}\). An x-value will correspond to only one y-value. Hence, it meets the definition of a function.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are foundational in mathematics and help us describe relationships using letters and numbers. When we see an equation like \(x^2 + y - 16 = 0\), it tells us how two variables, \(x\) and \(y\), relate to each other.

In algebraic terms, an equation is a statement that asserts the equality between two expressions. Here are some key aspects to remember:

• An equation can have one or more variables.
• It can be linear, quadratic, or of higher degree, depending on the highest power of the variable. For example, \(x^2\) indicates a quadratic equation.
• Solving an equation typically involves finding the value(s) of the variable(s) that make the equation true.

Equations are the language of algebra, allowing us to describe patterns, solve problems, and understand mathematical relationships.

Defining Functions

A function in mathematics sets up a relationship between input and output values. It assigns exactly one output to each input.

Given the equation \(y = 16 - x^2\), we want to check if it defines \(y\) as a function of \(x\).

• For a relationship to be a function, every input must have a unique output.
• If you can express \(y\) in terms of \(x\) such that for each \(x\), there is only one \(y\), it confirms \(y\) is a function of \(x\).
• The equation \(y = 16 - x^2\) does this, so \(y\) is indeed a function of \(x\).

Functions are powerful tools in algebra and calculus. They let us model real-world phenomena mathematically and perform various operations like differentiation and integration.

Solving Equations

Solving an equation means finding the value of the variable that makes the equation true. Here’s how we do it step by step based on our original exercise:

• Start by writing down the equation: \(x^2 + y - 16 = 0\).
• Next, rearrange the equation to solve for \(y\). In this case, isolate \(y\) on one side: \(y = 16 - x^2\).
• Check if the structure of the equation follows the criteria of a function. Here, for every \(x\), there's only one \(y\), confirming it is a function.

When solving equations, always aim to isolate the variable of interest by using algebraic operations like addition, subtraction, multiplication, or division.

Solving equations is critical in everything from engineering to everyday problem-solving, enabling us to find unknown values and make informed decisions.