Problem 12 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Yes, the function is one-to-one.

Step 1: Understanding the One-to-One Function

A function is one-to-one if every element of the range is mapped to by exactly one element of the domain. In simpler terms, different inputs should result in different outputs for the function.

Step 2: Algebraic Test for One-to-One Function

To determine if the function \( f(x) = 3x - 2 \) is one-to-one, we assume that \( f(x_1) = f(x_2) \) for different values \( x_1 \) and \( x_2 \), and check if \( x_1 \) must equal \( x_2 \).

Step 3: Set up the Equation

Assume that \( f(x_1) = f(x_2) \). This means\( 3x_1 - 2 = 3x_2 - 2 \).

Step 4: Simplify the Equation

Cancel out the constants from both sides:\( 3x_1 - 2 + 2 = 3x_2 - 2 + 2 \).This simplifies to:\( 3x_1 = 3x_2 \).

Step 5: Solve for Variables

Divide both sides by 3:\( x_1 = x_2 \).This result demonstrates that different inputs will not have the same output, which confirms the function is one-to-one.

Key Concepts

Function Analysis

When analyzing the function \( f(x) = 3x - 2 \) to determine if it is one-to-one, you need to understand what this means. A function is considered one-to-one if every output value corresponds to one and only one input value. This is important because it assures us that the function has an inverse. Here’s how it works:

• If you input different values into the function, the outputs must also be different.
• By knowing this property, we can avoid functions that overlap or repeat outputs for different inputs.

In practical terms, this means inspecting the function's behavior as we evaluate it at distinct points. A one-to-one function makes sure each output is unique and directly traceable back to a single input.

Algebraic Test

One effective way to confirm if a function is one-to-one is through the algebraic test. Let's look at our specific function, \( f(x) = 3x - 2 \), and verify its one-to-one nature using algebra.To do this:

1. Assume that two elements of the domain, say \( x_1 \) and \( x_2 \), are mapped to the same element in the range: \( f(x_1) = f(x_2) \).
2. This gives us the equation: \( 3x_1 - 2 = 3x_2 - 2 \).
3. By simplifying, we add 2 to both sides to obtain \( 3x_1 = 3x_2 \).
4. Then, divide both sides by 3 to get \( x_1 = x_2 \).

The final step confirms our hypothesis: if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \). Hence, \( f(x) = 3x - 2 \) passes the algebraic test for being one-to-one.

Domain and Range

Understanding the domain and range of a function is fundamental, especially when analyzing whether it is one-to-one. The domain of a function is the complete set of possible values of \( x \) you can input into it to get a real number. For \( f(x) = 3x - 2 \), the domain is all real numbers because any real number can be multiplied by 3 and then reduced by 2 without restriction.On the other hand, the range is the set of all possible outputs of the function. Since the function involves a linear expression without bounds, every real number can be an output of \( f(x) \), meaning the range is also all real numbers.In simpler terms:

• Domain: All real numbers.
• Range: All real numbers.

By knowing the domain and range, we ensure that the one-to-one property can be properly evaluated, as we assure that the function maps inputs and outputs without repetition across the entire set of real numbers.