Problem 8 - Algebra for College Students

Question

Determine whether the function \(f\) is one-to-one. $$ f(x)=-3 x+4 $$

Step-by-Step Solution

Answer

The function \( f(x) = -3x + 4 \) is one-to-one.

Step 1: Understand the function

The function given is linear and is expressed as \( f(x) = -3x + 4 \). A linear function of the form \( f(x) = mx + b \) is one-to-one if the coefficient \( m \) is not zero.

Step 2: Check the slope

The slope of the function \( f(x) = -3x + 4 \) is \( -3 \). Since the slope \( -3 eq 0 \), our function has a consistent direction of increase or decrease.

Step 3: Apply the horizontal line test

A one-to-one function graph does not have any horizontal line intersecting it more than once. Since the function is linear with a non-zero slope, every horizontal line will intersect it exactly once.

Step 4: Conclude based on the test

Since every horizontal line intersects the linear function \( f(x) = -3x + 4 \) exactly once, it is a one-to-one function.

Key Concepts

Understanding Linear Functions

Linear functions are a fundamental type of function often represented in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. They create straight lines when graphed, making them straightforward to analyze.

• The slope \( m \) indicates the steepness or inclination of the line.
• The y-intercept \( b \) indicates where the line crosses the y-axis.

In the context of determining whether a function is one-to-one, linear functions are special. If the slope \( m \) is non-zero, the function can potentially be one-to-one. This means that for each unique \( x \) value, there is a unique \( y \) value, which is a basic requirement for a function to be one-to-one. Linear functions like \( f(x) = -3x + 4 \) have continuous predictability due to their straight path unless the slope is zero.

Analyzing the Slope

The slope is a crucial concept when considering whether a linear function is one-to-one. In the function \( f(x) = -3x + 4 \), the slope is \(-3\). But what does this mean?

• A slope of \(-3\) means the line decreases, going downwards as you move from left to right on a graph.
• A slope other than zero ensures that the function moves consistently in one direction, either up or down.

If the slope is zero, the function becomes a constant horizontal line, which would not be one-to-one because it yields the same \( y \) value for multiple \( x \) values. Hence, in checking if a function is one-to-one, verifying that the slope is non-zero simplifies the process greatly.

Using the Horizontal Line Test

The horizontal line test is a simple yet powerful tool used to determine if a function is one-to-one. It involves drawing horizontal lines through various points on a graph and checking how many times these lines intersect with the graph.

• If any horizontal line intersects the graph more than once, the function is not one-to-one.
• For a function like \( f(x) = -3x + 4 \), each horizontal line intersects the graph exactly once due to its non-zero slope.

This test stems from the definition of a one-to-one function: each output is paired with exactly one input. Linear functions with non-zero slopes naturally pass this test as their lines are straight and consistently trend in a single direction, thereby ensuring unique \( y \) values for unique \( x \) values. This is why the horizontal line test is both quick and reliable for confirming one-to-one status in linear functions.