Problem 8 - Algebra and Trigonometry

Question

Determine whether the function is one-to-one. \(f(x)=3 x-2\)

Step-by-Step Solution

Answer

The function \( f(x) = 3x - 2 \) is one-to-one.

Step 1: Understanding the Function

The function given is a linear function in the form of \( f(x) = 3x - 2 \). We need to determine if it is one-to-one, which means each output value \( f(x) \) is unique for every different input value \( x \).

Step 2: Define One-to-One Function

A function is one-to-one if and only if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \). This means that for the same function outputs, the inputs must be identical.

Step 3: Solve the One-to-One Condition

Assume \( f(x_1) = f(x_2) \). This translates into \( 3x_1 - 2 = 3x_2 - 2 \). Simplifying this equation, we cancel out the -2 from both sides, getting \( 3x_1 = 3x_2 \). Dividing both sides by 3 gives \( x_1 = x_2 \).

Step 4: Conclusion

Since \( x_1 = x_2 \) holds true for every scenario where \( f(x_1) = f(x_2) \), the function \( f(x) = 3x - 2 \) is one-to-one. This confirms there are no two different inputs producing the same output.

Key Concepts

Linear Function

A linear function is a mathematical expression that models a straight line when graphed on a coordinate plane. These functions are typically written in the form of \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In other words, the function will always produce a straight line, which indicates a constant rate of change. For example, in the function \( f(x) = 3x - 2 \), \( m = 3 \) represents the slope, showing that for every one unit increase in \( x \), the value of the function increases by 3. The y-intercept \( b = -2 \) points to where the line crosses the y-axis.

• Linear functions help in understanding relationships between two variables.
• They are crucial in both algebra and real-world applications, such as calculating distances, prices, and time.

Understanding these properties is fundamental when determining whether the function is one-to-one.

Unique Output

A function generates a unique output if each input corresponds to only one output. This characteristic is essential for a function to be categorized as one-to-one. In a linear function like \( f(x) = 3x - 2 \), uniqueness can be tested. Imagine having different \( x \) values; as long as the line is neither horizontal nor a constant, each \( x \) produces a distinct \( f(x) \).

If the function had the form \( y = c \), where \( c \) is a constant, it wouldn't be one-to-one because every value of \( x \) gives the same \( y \). Fortunately, for non-horizontal linear functions, this isn't the case.

• Unique output reinforces the function's reliability in predicting specific results from given inputs.
• It ensures no overlap in outputs, indicating one-to-one nature.

Identifying unique outputs guides us in determining the precise behavior of algebraic functions.

Function Properties

Understanding function properties is crucial when working with one-to-one functions. Let's explore what these properties are and how they apply to linear functions like \( f(x) = 3x - 2 \). Key properties include:

• Domain and Range: For this function, the domain and range are all real numbers because there are no restrictions on \( x \) in a linear equation.
• Continuity: Linear functions are continuous, meaning there are no breaks, jumps, or holes in the graph.
• Increasing/Decreasing Behavior: This specific function is strictly increasing since the slope \( m = 3 \) is positive.
• One-to-One Characteristic: A function is one-to-one when each \( x \) value has a unique \( y \) value, which also aligns with the strict increase due to the constant positive slope.

By understanding these properties, determining whether a function is one-to-one becomes straightforward.

Algebra Concepts

Algebra provides the tools to assess whether a function is one-to-one. Crucially, the concept relies on testing equality. The test involves assuming equality between two different inputs producing the same output and proving that those inputs must be identical. Here’s a breakdown using our function \( f(x) = 3x - 2 \):

- Suppose \( f(x_1) = f(x_2) \), which translates to \( 3x_1 - 2 = 3x_2 - 2 \)
- Solving further, we cancel \( -2 \) from both sides, leaving \( 3x_1 = 3x_2 \)
- Dividing by 3 gives \( x_1 = x_2 \)

This algebraic step confirms the function is indeed one-to-one. Understanding such algebra concepts not only helps in analyzing functions but also strengthens problem-solving skills in various mathematical scenarios. Using these principles allows us to ascertain function behaviors confidently in different contexts.