Problem 7 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function \(f(x) = -2x + 4\) is one-to-one.

Step 1: Define a One-to-One Function

A function is considered one-to-one if and only if every value of the function's output corresponds to exactly one input. This means that if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\). An alternate way to determine if a function is one-to-one is to check if its derivative is always positive or always negative (for differentiable functions).

Step 2: Differentiate the Function

Calculate the derivative of the function \(f(x) = -2x + 4\). Since it is a linear function, we can find the derivative by differentiating each term individually. The derivative of \(-2x\) is \(-2\), and the derivative of a constant \(4\) is 0. Thus, the derivative of the function is \(f'(x) = -2\).

Step 3: Analyze the Derivative

Since the derivative \(f'(x) = -2\) is constant and negative for all values of \(x\), the function is strictly decreasing. A function that is strictly increasing or strictly decreasing is a one-to-one function.

Step 4: Conclude

Based on the constant and negative derivative, \(f(x) = -2x + 4\) is a one-to-one function because it is strictly decreasing, meaning it passes the horizontal line test; no horizontal line will intersect the graph more than once.

Key Concepts

Linear Functions

Linear functions are perhaps the simplest type of functions one can encounter in mathematics. A linear function can be recognized by its standard form: \( f(x) = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept.
The slope \( m \) indicates the steepness and direction of the line:

• If \( m > 0 \), the line slopes upward; it is increasing.
• If \( m < 0 \), the line slopes downward; it is decreasing.
• If \( m = 0 \), the line is horizontal; it is constant.

The function \( f(x) = -2x + 4 \) is a classic linear function, where \( m = -2 \) and \( b = 4 \). The negative slope indicates that the function is decreasing. The y-intercept \( b = 4 \) shows that the line crosses the y-axis at the point (0, 4). With linear functions, what you observe is a straight line, which simplifies the analysis greatly due to consistent behavior.

Derivative

The derivative is a powerful tool in calculus that helps to understand how functions change. In more practical terms, the derivative of a function at any particular point provides the rate at which the function's value changes with respect to changes in the input.
For a linear function \( f(x) = mx + b \), the derivative is particularly simple because it is constant and equal to the slope \( m \). The resulting derivative tells us about the behavior of the function:

• If the derivative is positive, the function is increasing.
• If the derivative is negative, the function is decreasing.
• If the derivative is zero, the function is constant over its domain.

In the function \( f(x) = -2x + 4 \), the derivative \( f'(x) = -2 \) is constant and negative, predicting a uniformly decreasing trend across the entire graph.

Strictly Decreasing Function

A strictly decreasing function is one where, as the input increases, the output consistently decreases. This means that if you pick any two distinct inputs \( x_1 \) and \( x_2 \) such that \( x_1 < x_2 \), then \( f(x_1) > f(x_2) \).
By examining the derivative, we can determine if a function is strictly decreasing across its domain. If the derivative is negative at every point, the function falls into this category.
For \( f(x) = -2x + 4 \), the derivative \( f'(x) = -2 \) is constant and negative, indicating the function is strictly decreasing. This characteristic ensures the function is one-to-one, as no two distinct inputs yield the same output. Thus, a strictly decreasing or increasing function automatically passes the horizontal line test—proof that each output is unique to a single input.