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Chapter 5: Problem 2

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=-5 x+2$$

Short Answer

Expert verified
The function \( f(x) = -5x + 2 \) is one-to-one.

Step by step solution

01

Identify Function Type

The function given is a linear function of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this case, \( m = -5 \) and \( b = 2 \). Linear functions are inherently either one-to-one or many-to-one based on their slope.
02

Analyze Slope

Observation tells us that a linear function with a non-zero slope is one-to-one. This is because the function passes the horizontal line test: every horizontal line intersects the graph at most once. Here the slope \( -5 \) is non-zero, therefore, \( f(x) = -5x + 2 \) is one-to-one.
03

Conclusion Verification

Verify the conclusion by checking that if \( f(a) = f(b) \), it implies \( a = b \). For \( f(x) = -5x + 2 \), if \( -5a + 2 = -5b + 2 \), removing the common terms results in \( a = b \), confirming the function is one-to-one.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

One-to-One Functions
A **One-to-One Function** is a type of function where each input has a unique output, and each output is associated with a single input. This means no horizontal line will touch the graph of a one-to-one function more than once.
  • To determine if a function is one-to-one, you can use the *horizontal line test*. If any horizontal line touches the graph more than once, the function is not one-to-one.
  • In the case of linear functions, such as the example given by the equation \( f(x) = -5x + 2 \), these functions are typically one-to-one if their slope \( m \) is not zero. This is because the slope determines the angle at which the line rises or falls, which affects how it interacts with horizontal lines.
Understanding one-to-one functions is key in mathematics because they ensure that each value in the range and domain map to each other precisely, which is crucial in inverse operations.
Function Analysis
**Function Analysis** involves examining the properties and characteristics of a given function to understand its behavior better.
  • To analyze a function, you first need to identify its form and components. For linear functions like \( f(x) = mx + b \), recognize the constants \( m \) (slope) and \( b \) (y-intercept).
  • The behavior of the function is largely determined by the slope \( m \). A positive slope indicates the function graph rises as it moves from left to right, while a negative slope, such as \( -5 \) in our case, means the graph falls.
  • The y-intercept \( b \) indicates where the function crosses the y-axis. For \( f(x) = -5x + 2 \), the line crosses at \( y = 2 \).
Function analysis is an essential skill as it helps predict how changing parameters alter the graph's appearance and functionality.
Slope of a Line
The **Slope of a Line** is a crucial concept in linear functions, represented by the coefficient \( m \) in the equation \( y = mx + b \). It dictates how steep the line is and the direction it goes.
  • For the equation \( f(x) = -5x + 2 \), the slope is \( -5 \). This slope tells us two things: first, the line falls as you move from left to right (a negative slope). Second, it falls steeply, as a larger absolute value implies a steeper slope.
  • The slope can be calculated using two points on the line: \((x_1, y_1)\) and \((x_2, y_2)\). The formula is: \( m = \frac{y_2-y_1}{x_2-x_1} \).
Understanding the slope of a line helps not only in determining the angle of the line but also in analyzing the relationship between variables in linear equations.

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