Problem 23 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is one-to-one. You may want to graph \(y=f(x)\) and apply the horizontal line test. $$ f(x)=2 x-7 $$

Step-by-Step Solution

Answer

Yes, the function \(f(x) = 2x - 7\) is one-to-one.

Step 1: Define One-to-One Function

A function is one-to-one if every horizontal line intersects the graph of the function at most once. This means that each output value of the function corresponds to exactly one input value.

Step 2: Analyze the Function Type

The given function is a linear function of the form \(f(x) = mx + c\), where in this case, \(m = 2\) and \(c = -7\). Linear functions with non-zero slopes are always one-to-one because they are either strictly increasing or strictly decreasing.

Step 3: Apply the Horizontal Line Test

Since \(f(x) = 2x - 7\) is a linear function with a non-zero slope (\(m = 2\)), it is strictly increasing. This means any horizontal line will intersect the graph at most once, confirming that the function is one-to-one.

Step 4: Conclusion from Graph

Although graphing is not necessary in this linear case, graphing \(f(x) = 2x - 7\) would show a straight line with positive slope, reinforcing that the function is one-to-one.

Key Concepts

Horizontal Line Test

In mathematics, determining if a function is "one-to-one" can often be accomplished using the horizontal line test. This simple yet powerful test involves imagining or drawing horizontal lines across the graph of the function. Key aspects include:

• If any horizontal line intersects the graph more than once, the function is not one-to-one.
• If no horizontal line intersects more than once, the function passes this test and is one-to-one.

Given this concept, functions like linear ones can be quickly evaluated. Specifically, a linear function with a non-zero slope will always be one-to-one. Thus, the test assures uniqueness of outputs for inputs. This is crucial for understanding functions' behavior.

Linear Function

The concept of a linear function is fundamental in math, characterized by a constant slope and represented as \( f(x) = mx + c \). Important points to remember include:

• "m" is the slope of the line, determining its steepness and direction.
• "c" denotes the y-intercept, where the line crosses the y-axis.

For the function \( f(x) = 2x - 7 \):

• The slope \( m = 2 \) indicates a constant upward trend.
• The y-intercept \( c = -7 \) tells us the starting point on the y-axis.

Linear functions with non-zero slopes, like this one, are always one-to-one and do not overlap horizontally, ensuring a unique output for every input value.

Graphing Functions

Graphing functions is an intuitive way to visualize how functions behave. It helps in quickly identifying properties like one-to-one nature. Here’s how graphing is helpful:

• Provides a clear visual representation of the slope and intercept.
• Allows for easy application of tests like the horizontal line test.

In the case of \( f(x) = 2x - 7 \), you would graph a line starting at \( (0, -7) \) that moves upward at a consistent angle. This visual confirms:

• The positive slope suggests a strictly increasing function.
• Any horizontal line would only meet the function graph once.

Graphing is not always required for determining one-to-one status, especially for linear functions. However, it serves as an excellent tool for enhancing understanding and verifying algebraic conclusions.