Problem 11 - Algebra and Trigonometry

Question

Determine whether the function is one-to-one. $$ f(x)=-2 x+4 $$

Step-by-Step Solution

Answer

The function is one-to-one because it is a linear function with a non-zero slope of -2.

Step 1: Define a One-to-One Function

A function is one-to-one if and only if each value of the function’s output corresponds to exactly one input value. In other words, no two distinct inputs have the same output value.

Step 2: Determine the Type of Function

Examine the given function: \[ f(x) = -2x + 4 \]This is a linear function because the equation is of the form \( y = mx + b \), where \( m = -2 \). Linear functions are one-to-one if their slope \( m \) is non-zero.

Step 3: Check for a Non-Zero Slope

For the linear function \( f(x) = -2x + 4 \), the slope \( m \) is -2. Since the slope \( -2 eq 0 \), the function is one-to-one.

Key Concepts

Linear Function

A linear function is one of the simplest and most foundational concepts in algebra. To identify a linear function, look for equations that fit the form \( y = mx + b \). Here:

• \( y \) is the output.
• \( x \) is the input.
• \( m \) represents the slope of the line.
• \( b \) is the y-intercept where the line crosses the y-axis.

Linear functions graph as straight lines in the coordinate plane. Each value of \( x \) produces a single, unique value of \( y \), making these functions predictable and easy to analyze.
In the context of one-to-one functions, linear functions are particularly interesting. Not all are one-to-one, but they can be under certain conditions, which we'll explore further! Understanding the basics of linear functions is critical for deeper insights into more complex topics like calculus and differential equations.

Non-Zero Slope

The slope \( m \) in a linear function determines the direction and steepness of the line. A key point is that a function is one-to-one if its graph is a line that doesn’t run horizontally, signified by a non-zero slope.
In our example function \( f(x) = -2x + 4 \), the slope \( m = -2 \) is unmistakably non-zero. This non-zero slope means:

• The line is not horizontal.
• Each input \( x \) maps to a unique output \( y \).
• The function does not overlap itself, ensuring it’s one-to-one.

A positive slope indicates an upward direction as x increases, while a negative slope indicates a downward movement.
Thus, by checking the slope, you can confidently determine whether a linear function is one-to-one, which helps in broader analysis like determining inverse functions.

Function Analysis

Function analysis involves examining the properties of a function to understand its behavior more deeply. For the linear function \( f(x) = -2x + 4 \), we performed an analysis to determine it is one-to-one by confirming its non-zero slope.
This function analysis process can be broken down simply:

• Identify the function type: Recognizing a linear function helps to apply the appropriate properties and formulas.
• Check the slope: A non-zero slope confirms unique mapping of every input to a single output.
• Conclude the one-to-one nature: Demonstrates no two inputs produce the same output, confirming the function's distinct mapping capability.

Performing careful function analysis is vital for solving more complex mathematical problems and deducting meaningful insights from equations.
Understanding these principles aids in tasks like finding the inverse of a function or predicting the behavior of data described by that function.