Problem 35 - Elementary and Intermediate Algebra

Question

Determine whether each equation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x .\) See Example 3. $$ y=2 x+3 $$

Step-by-Step Solution

Answer

Yes, \(y = 2x + 3\) defines \(y\) to be a function of \(x\).

Step 1: Assess Linear Function Form

The given equation is \(y = 2x + 3\). This is a linear equation in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. A linear equation in this form typically defines \(y\) as a function of \(x\).

Step 2: Identify Function Definition

A function is defined such that each input (\(x\) value) corresponds to exactly one output (\(y\) value). In the equation \(y = 2x + 3\), for any given \(x\) value, there is one and only one result for \(y\).

Step 3: Verify Unique Output for Input

Substitute different \(x\) values into the equation to observe the output. For example, when \(x = 1\), \(y = 2(1) + 3 = 5\). When \(x = 2\), \(y = 2(2) + 3 = 7\). In each case, a unique \(x\) gives a unique \(y\).

Step 4: Conclude Function Status

Since every \(x\) value corresponds to exactly one \(y\) value, \(ye 2x+3\) defines \(y\) to be a function of \(x\).

Key Concepts

Linear Equations

Linear equations are fundamental in the world of mathematics, describing a straight line on a graph. They are equations where each term is either a constant or the product of a constant and a single variable. A typical example of a linear equation is given in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

Linear equations are significant because they establish a linear relationship between dependent and independent variables. In simpler terms, as you change \( x \), \( y \) changes in a consistent, predictable manner; this is perfectly represented by a straight line.

Some essential characteristics of linear equations include:

• They have constant rates of change or slopes.
• The graph of a linear equation is always a straight line.
• They have no exponents higher than one for any variable.

In the exercise provided, the equation \( y = 2x + 3 \) demonstrates these properties and defines a straight line, indicating it is a linear equation.

Slope-Intercept Form

The slope-intercept form, designated mathematically as \( y = mx + b \), is a particular way of expressing the equation of a line. Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, where the line crosses the y-axis.

This form is valuable because it easily presents the slope, which tells us both the direction and steepness of the line. If \( m \) is positive, the line ascends as it moves right; if \( m \) is negative, it descends. The y-intercept \( b \) indicates where the line crosses the y-axis.

Knowing the slope and y-intercept gives us a powerful insight into how a line behaves, making it easy to plot the line graphically or analyze its properties. In our given equation \( y = 2x + 3 \), the slope is 2, implying the line rises steeply, and the y-intercept is 3, indicating it crosses the y-axis at point (0, 3). This confirms the equation is expressed in slope-intercept form.

Input-Output Relationship

The concept of input-output relationships is crucial in understanding functions. A function is a special kind of relation where each input corresponds to exactly one output. In mathematical terms, this means for every value of \( x \), there is a unique value of \( y \).

The equation \( y = 2x + 3 \) models this relationship perfectly. As you choose any \( x \) value (input), you compute \( y \) (output) using the given equation. For instance, if \( x = 4 \), then \( y = 2(4) + 3 = 11 \).

Let's summarize the key points about input-output relationships:

• Each input has a single output, ensuring one-to-one correspondence.
• In our case, substituting different inputs into the function results in different outputs, verifying the function's integrity.

Therefore, our equation \( y = 2x + 3 \) establishes that every \( x \) maps to exactly one \( y \), satisfying the condition of a function.