Problem 5 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=3 x-7 $$

Step-by-Step Solution

Answer

The function \( f(x) = 3x - 7 \) is one-to-one.

Step 1: Understanding One-to-One Functions

First, recall the definition of a one-to-one function. A function is one-to-one if and only if, for every pair of distinct inputs, the outputs are also distinct. In mathematical terms, if \( f(x_1) = f(x_2) \), then it must be the case that \( x_1 = x_2 \). We will use this definition to determine if the given function is one-to-one.

Step 2: Setting Up the Equation

Consider the given function \( f(x) = 3x - 7 \). Start by assuming that \( f(x_1) = f(x_2) \), which gives us the equation: \[ 3x_1 - 7 = 3x_2 - 7 \] We need to prove that this implies \( x_1 = x_2 \).

Step 3: Simplifying the Equation

Eliminate the constant \(-7\) from both sides of the equation: \[ 3x_1 = 3x_2 \] Next, divide both sides by 3: \[ x_1 = x_2 \] This shows that the function passes the one-to-one test since identical outputs imply the inputs are identical.

Step 4: Conclusion

Since for any \( x_1 \) and \( x_2 \), if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \), the function \( f(x) = 3x - 7 \) is indeed one-to-one.

Key Concepts

Function Properties

Functions are fundamental building blocks in mathematics that have certain properties and characteristics. Knowing these properties helps us understand how functions behave. A function essentially maps each element from a set, called the domain, to a unique element in another set, called the codomain.

Here are key properties of functions that are vital:

• **Domain and Range:** The domain is the set of all possible input values, while the range is the set of all possible output values.
• **Continuity:** A function is continuous if there are no breaks or jumps in its graph.
• **Injective or One-to-One:** A function is one-to-one if distinct inputs always produce distinct outputs.
  
  Understanding these properties allows us to predict and examine the behavior of specific functions. This helps in identifying special types such as one-to-one functions, which are essential in various mathematical analyses and applications.

Linear Functions

Linear functions are one of the simplest types of functions. They take the form of a straight line when graphed. The general form of a linear function is given by: \[ f(x) = ax + b \]where:

• \(a\) is the slope of the line, which indicates the steepness and the direction of the line.
• \(b\) is the y-intercept, indicating where the line crosses the y-axis.

Linear functions are vital because they model a constant rate of change.

In the context of the exercise, the function \( f(x) = 3x - 7 \) has:

• a slope \(a = 3\), which means for every unit increase in \(x\), \(f(x)\) increases by 3 units.
• a y-intercept \(b = -7\), meaning the line crosses the y-axis at -7.

Linear functions are straightforward to analyze, especially when determining properties such as whether they are one-to-one.

Injective Functions

Injective functions, also known as one-to-one functions, have a straightforward and crucial property. They ensure that each element of the function's domain maps to a unique element in the codomain. This means no two distinct inputs will yield the same output.

Mathematically, a function \( f(x) \) is injective if whenever \( f(x_1) = f(x_2) \), it follows that \( x_1 = x_2 \). In simpler terms, different inputs always lead to different outputs.

To determine if a linear function is injective, we can use a simple method. Consider two inputs \( x_1 \) and \( x_2 \):

• Assume \( f(x_1) = f(x_2) \).
• For the linear function \( f(x) = ax + b \), derive the equation: \( ax_1 + b = ax_2 + b \).
• Simplify to \( ax_1 = ax_2 \), leading to \( x_1 = x_2 \).

This process confirms that the function is injective if \( a eq 0 \).

In the exercise, the linear function \( f(x) = 3x - 7 \) has \( a = 3 \), proving it is indeed one-to-one, confirming injectivity.