Problem 36 - 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. $$ y=4 x-1 $$

Step-by-Step Solution

Answer

The equation \(y = 4x - 1\) defines \(y\) as a function of \(x\).

Step 1: Understanding the Definition of a Function

A relationship between two variables, typically called \(x\) and \(y\), is called a function if each input value (\(x\)) maps to exactly one output value (\(y\)). We'll examine the given equation to see if it meets this criteria.

Step 2: Analyze the Given Equation

The given equation is \(y = 4x - 1\). This equation is expressed in the form \(y = mx + b\), where \(m\) and \(b\) are constants. This is the standard form of a linear equation, which generally represents a straight line.

Step 3: Check One-to-One Correspondence

For \(y = 4x - 1\), each value of \(x\) determines exactly one value of \(y\). For instance, if \(x = 2\), then \(y = 4(2) - 1 = 7\). As you see, there is no other value of \(y\) for \(x = 2\). Similarly, for any value of \(x\), substituting back into the linear equation results in exactly one value of \(y\).

Step 4: Conclusion

Since every \(x\) produces exactly one \(y\) in the equation \(y = 4x - 1\), the equation defines \(y\) as a function of \(x\). There are no cases where a single \(x\) value corresponds to more than one \(y\) value.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and describe a straight line when plotted on a graph. In simple terms, a linear equation in two variables typically takes the form \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. Slope represents how steep the line is, while the y-intercept is where the line crosses the y-axis.

Understanding linear equations is crucial because they form the basis for more complex mathematical concepts. They're easy to work with due to their straightforward nature and predictability.

• They represent constant rates of change.
• They graph as straight lines.
• They have no exponents higher than one on any variable.

The linear equation \( y = 4x - 1 \) fits this model perfectly with a slope \( m = 4 \) and intercept \( b = -1 \). This means every increase in \( x \) by one unit changes \( y \) by four units, maintaining a consistent pattern.

Function Definition

The concept of a function is about mapping each element from a set of inputs to exactly one output. Think of it like a vending machine: for every option (input) you select, you receive one product (output).

A mathematical function is denoted as \( f(x) \), where \( f \) represents the function and \( x \) is an input from the function's domain. This domain is the set of all possible inputs that the function can accept. The resulting outputs form the range.

• A function must associate each input with only one output.
• Repeat inputs yielding different outputs violate the definition of a function.
• Functions can be linear, like \( y = 4x - 1 \), or more complex. What matters is the one-to-one dependency.

In the equation \( y = 4x - 1 \), every \( x \) corresponds to exactly one \( y \). This shows it is indeed a function.

One-to-One Correspondence

The idea of one-to-one correspondence means for every input, there is a unique output. This is what makes a function reliable and predictable.

To check if a function is one-to-one, we observe the uniqueness of output values for inputs. If a single input value can map to two different outputs, the function fails the one-to-one test and is not a function in the formal sense.

• For \( y = 4x - 1 \), if \( x = 2 \), then \( y = 7 \) as seen from the equation.
• There cannot be another \( y \) value when \( x = 2 \).
• Thus, no input value \( x \) results in different outputs, confirming one-to-one correspondence.

This property of \( y = 4x - 1 \) ensures it functions smoothly as a mapping from \( x \) to \( y \), retaining the essence of a linear function.