Problem 11 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

Yes, \(y = 16 - x\) clearly states that the equation defines \(y\) as a function of \(x\). For every input of \(x\), there is exactly one output of \(y\), which is the characteristic of a function.

Step 1: Rearrange the given equation

Rearrange the given equation to isolate \(y\) on one side. This can be done by subtracting \(x\) from both sides. This gives: \(y = 16 - x\)

Step 2: Check the Function Rule

In the equation \(y = 16 - x\), for any given value of \(x\), there is exactly one corresponding value of \(y\). This fulfills the requirement for \(y\) to be a function of \(x\). Therefore, the given equation defines \(y\) as a function of \(x\).

Key Concepts

Equation Manipulation

Equation manipulation involves rearranging an equation to make it easier to work with or to extract specific information. In the context of determining functions, manipulating equations can help isolate variables and clarify relationships between them.

The first step in the given exercise involves rearranging the equation \(x + y = 16\). To identify \(y\) as a function of \(x\), we aim to isolate \(y\) on one side of the equation. By subtracting \(x\) from both sides, we transform the equation into \(y = 16 - x\).

This manipulation is crucial because it allows us to directly see how changes in \(x\) affect \(y\). Such clarity is essential in understanding functions, making equation manipulation an invaluable skill in mathematics.

Linear Equations

Linear equations form the basis of many mathematical functions and are characterized by a constant rate of change. The equation \(y = 16 - x\) is a linear equation because it can be written in the form \(y = mx + b\), where \(m\) and \(b\) are constants.

In this equation:

• \(m = -1\), which is the slope or the rate at which \(y\) changes concerning \(x\).
• \(b = 16\), the y-intercept, representing the value of \(y\) when \(x = 0\).

These elements define a line on a graph, and any linear equation can be graphically represented by a straight line. Linear equations are fundamental in algebra and serve as a stepping stone to understand more complex functions.

Recognizing the form of a linear equation aids in quickly assessing the relationship between variables, such as identifying \(y\) as a function of \(x\).

Function Rule

A function rule describes how one quantity depends on another. Specifically, a relation is defined to be a function if, for every input, there is exactly one output. In simple terms, this means that each value of \(x\) produces a unique value of \(y\).

In the context of our rearranged equation \(y = 16 - x\), the function rule states:

• For any chosen value of \(x\), there is a corresponding unique value of \(y\).
• This consistency satisfies the criteria for \(y\) to be a function of \(x\).

Understanding function rules is fundamental in mathematics because they ensure predictable and repeatable outputs from given inputs.

Once you properly manipulate an equation and determine it fits a function’s definition, you can confidently express and utilize \(y\) in terms of \(x\) as a function.