Problem 19 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand One-to-One Functions

A function is one-to-one if for every pair of distinct inputs, the outputs are distinct. In other words, if \( f(a) = f(b) \) implies that \( a = b \). This means no two different inputs map to the same output.

Step 2: Test the Function Algebraically

To check if \( f(x) = 2x \) is one-to-one, assume \( f(a) = f(b) \). Then, substitute to get \( 2a = 2b \). Divide both sides by 2, resulting in \( a = b \). This shows that if two outputs are equal, then the inputs must also be equal.

Step 3: Conclude Based on the Algebraic Test

Since \( a = b \) is the result of \( f(a) = f(b) \), this confirms that different inputs always yield different outputs. Hence, the function is one-to-one.

Key Concepts

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. In the context of functions, algebra allows us to explore and verify the properties of different types of functions. When examining whether a function is one-to-one, algebraic manipulation is key. We utilize equations to demonstrate how inputs relate to outputs. For instance, considering the function \( f(x) = 2x \), we test its nature by seeing if \( f(a) = f(b) \) necessitates \( a = b \). This involves setting up the equation \( 2a = 2b \) and applying standard algebraic steps like dividing both sides by 2 to simplify to \( a = b \). These steps show that algebra is instrumental in confirming function properties, particularly for establishing the one-to-one nature of a function.

Function Properties

Understanding function properties is essential to mastering mathematical functions and their classifications, such as whether a function is one-to-one.A one-to-one function, also known as an injective function, requires that each unique input corresponds to a unique output. For example:

• If \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \). This property ensures that no two different inputs can map to the same output value.
• Using the function \( f(x) = 2x \), we apply this by checking if \( f(a) = f(b) \). As we've derived, this leads to \( a = b \), confirming the function's injectiveness.

Ensuring that functions are one-to-one is critical, especially in contexts where reversibility is required, such as finding function inverses.

Mathematical Proof

Mathematical proof is a logical process used to establish the truth of mathematical statements. In determining if a function like \( f(x) = 2x \) is one-to-one, a proof involves demonstrating that the function meets the criteria for injectiveness. Here's a breakdown of a simple proof of one-to-one-ness:

• Assume \( f(a) = f(b) \) for a given function.
• Substitute the function expression: \( 2a = 2b \).
• Divide both sides of the equation by 2 to simplify it to \( a = b \).
• This simplification shows that only the same inputs can produce the same outputs, thus proving the function is one-to-one.

Proofs provide a structured methodology to verify and validate mathematical properties, ensuring clarity and rigor in conclusions.