Problem 20 - Algebra and Trigonometry

Question

Determine whether the function is one-to-one. $$ f(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

The function \(f(x) = \frac{1}{x}\) is one-to-one.

Step 1: Understand the Definition of a One-to-One Function

A function is one-to-one (injective) if and only if for every pair of different inputs \(x_1\) and \(x_2\), the outputs are different: \(f(x_1) eq f(x_2)\). Essentially, different inputs must map to different outputs.

Step 2: Identify the Domain of the Function

The function \(f(x) = \frac{1}{x}\) is defined for all real numbers except \(x = 0\). Therefore, the domain of \(f(x)\) is \(x \in \mathbb{R}, x eq 0\).

Step 3: Test the One-to-One Condition

Assume \(f(x_1) = f(x_2)\) and solve for \(x_1\) and \(x_2\) to check if they must be equal: \[\frac{1}{x_1} = \frac{1}{x_2}\]Cross-multiply to obtain \(x_2 = x_1\). Thus, if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\). This confirms that the function is one-to-one.

Key Concepts

Injective Function

An injective function, also commonly known as a one-to-one function, is a type of mapping where each element of the function's domain corresponds to a unique element in its codomain. This means that no two different inputs can produce the same output.

For a function to qualify as injective, we use a specific mathematical condition: for any two distinct numbers, say \(x_1\) and \(x_2\), in the domain of the function, it must hold that \(f(x_1) eq f(x_2)\). If this condition is satisfied, then it's clear that each output is distinct for distinct inputs.

To determine if a function is injective, you can take two approaches:

• Algebraically prove that assuming \(f(x_1) = f(x_2)\) leads to \(x_1 = x_2\).
• Graphically by checking if any horizontal line cuts the graph in more than one point. If no horizontal line does, the function is injective.

Understanding the injective property is crucial in various areas of mathematics since it helps in assuring the full mapping of elements without any overlaps.

Domain of a Function

When dealing with functions, defining the domain is key to understanding where the function is applicable or defined. The domain of a function is essentially the complete set of possible input values (usually represented as \(x\)) for which the function is defined.

For example, consider the function \(f(x) = \frac{1}{x}\). Here, the function includes division by \(x\). Division by zero is undefined in mathematics because it does not lead to a finite number. Therefore, the input \(x = 0\) must be excluded from the domain.

In this specific case, the domain for \(f(x) = \frac{1}{x}\) is all real numbers except zero:

• \(x \in \mathbb{R}, x eq 0\)

This specification ensures that the function operates correctly and avoids any undefined operations. Accurately determining the domain is vital not only for proper function usage but also for understanding its limitations and behavior.

Mathematical Proof

Mathematical proofs play a crucial role in confirming various properties of functions, such as whether they are injective. In the case of determining if \(f(x) = \frac{1}{x}\) is one-to-one, a proof involves demonstrating that the function satisfies the injective condition.

To prove that this function is injective, we start by assuming the outputs of the function for two inputs are the same: \(f(x_1) = f(x_2)\). For our function, this translates to:

• \(\frac{1}{x_1} = \frac{1}{x_2}\)

Using algebraic manipulation, we can find that \(x_1 = x_2\) by cross-multiplying:

• \(x_2 \cdot \frac{1}{x_2} = x_1 \cdot \frac{1}{x_1}\)
• Which simplifies to \(x_2 = x_1\)

This step is crucial as it directly verifies that different inputs result in different outputs, adhering to the injective condition.

The elegance of mathematical proofs lies in their ability to universally validate conditions and solve for uncertainties. They build confidence in mathematical conclusions by providing logical and structured reasoning.