Problem 89 - College Algebra with Modeling and Visualization

Question

Determine if \(y\) is a function of \(x\). $$ x+y=2 $$

Step-by-Step Solution

Answer

Yes, \(y\) is a function of \(x\) as it gives a unique output for each input.

Step 1: Understand the Definition of a Function

For \(y\) to be a function of \(x\), each value of \(x\) must map to exactly one unique value of \(y\). This means for every input \(x\), there can be only one output \(y\).

Step 2: Manipulate the Equation to Isolate y

The given equation is \( x + y = 2 \). To isolate \(y\), subtract \(x\) from both sides of the equation.\[y = 2 - x\]

Step 3: Determine if y is a Function of x

The expression \( y = 2 - x \) is a linear equation. For each value of \(x\), there is exactly one corresponding value of \(y\). This satisfies the condition for a function, so \(y\) is indeed a function of \(x\).

Key Concepts

Understanding Linear Equations

Linear equations are a fundamental concept in algebra, representing equations where variables have exponents of 1. These equations are characterized by a consistent rate of change and their graphical representation is a straight line. A basic form of a linear equation is given by:

• For a single variable:
  \( ax + b = 0 \)
• For two variables, such as in our example, it is often written as:
  \( y = mx + c \)

In this format, \( m \) represents the slope and \( c \) the y-intercept, giving us the ability to quickly visualize the line on a graph. Linear equations are popular because they are simple to solve and understand, forming the basis of more complex algebraic concepts.

Definition of a Function

A function connects each element of a set to a unique element of another set, maintaining a relationship where every input has exactly one output. In the context of mathematics, especially algebra, you encounter this relationship when determining if an expression is a function. For an equation like \( y = 2 - x \):

• The variable \( x \) represents the input.
• The variable \( y \) is dependent on \( x \), making it the output.

For \( y \) to be a function of \( x \), each value you substitute for \( x \) should yield a unique value for \( y \). Thus, linear equations like \( y = 2 - x \) naturally fit the definition of a function, owing to their straightforward structure and predictable output for every potential input.

Algebraic Manipulation Essentials

Algebraic manipulation involves rearranging and simplifying equations to make them more understandable and easier to solve. When determining if \( y \) is a function of \( x \), you'll often need to manipulate the original equation to isolate \( y \).For example, given the equation \( x + y = 2 \):

• Subtract \( x \) from both sides to isolate \( y \):

\[ y = 2 - x \]By performing this algebraic step, you transform the original equation into a form where \( y \) is explicitly expressed as a function of \( x \). This practice not only helps in identifying functions but also in preparing the equation for further operations such as graphing or solving additional problems.