Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

The function \( f(x)=\frac{1}{x^2} \) is not one-to-one.

Step 1: Understand One-To-One Function

A function is one-to-one (injective) if and only if different inputs result in different outputs. In mathematical terms, for a function \( f \), it should satisfy that for every pair of distinct inputs \( a \) and \( b \), \( f(a) eq f(b) \). For this exercise, we need to determine if \( f(x)=\frac{1}{x^2} \) is an injective function.

Step 2: Consider Function Behavior

Examine the function \( f(x)=\frac{1}{x^2} \). Notice that \( x^2 \) is always positive for real numbers except when \( x=0 \). Therefore, \( \frac{1}{x^2} \) will also always be positive. This suggests that different values of \( x \) could potentially give the same result.

Step 3: Find Counter-Example

Find two different inputs \( a \) and \( b \) where \( f(a)=f(b) \). Set the function equal for two distinct inputs: \( \frac{1}{a^2} = \frac{1}{b^2} \). This simplifies to \( a^2 = b^2 \), which implies \( a = b \) or \( a = -b \). Therefore, distinct inputs such as \( 1 \) and \( -1 \) give the same output \( f(1) = f(-1) = 1 \).

Step 4: Conclude Function Is Not One-To-One

Since we found a counter-example where \( f(x) \) gives the same output for two different inputs (e.g., \( 1 \) and \( -1 \)), it is clear that the function \( f(x)=\frac{1}{x^2} \) is not one-to-one.

Key Concepts

Injective Function

An injective function, also known as a one-to-one function, is where each output is formed by exactly one distinct input. This means that no two different inputs map to the same output. To determine if a function is injective, we need to check if different inputs always lead to different outputs.

Consider any two distinct inputs, say \( a \) and \( b \). For a function \( f \) to be injective, it is required that \( f(a) eq f(b) \). This property is crucial because it ensures that the function maps every element of the domain to a unique element in the co-domain, thus preserving distinction of the inputs through the function.

Function Behavior

The behavior of a function provides insight into how it maps inputs to outputs. In the exercise, we consider the function \( f(x)=\frac{1}{x^2} \).

This function's behavior is heavily influenced by its formula. Since \( x^2 \) is always positive (except at zero where it's undefined), the reciprocal \( \frac{1}{x^2} \) is also positive. This positivity remains consistent across the domain, yet this uniform behavior implies that different inputs might lead to similar or identical outputs.

• The function is undefined at \( x = 0 \), limiting the domain.
• Every positive and negative input |x| share the same output, indicating potential non-injectivity.

Counter-Example

To establish that a function is not injective, finding a counter-example is effective. This involves identifying two distinct inputs that produce the same output.

For the function \( f(x)=\frac{1}{x^2} \), consider inputs \( 1 \) and \( -1 \). Calculating their outputs:

• \( f(1) = \frac{1}{1^2} = 1 \)
• \( f(-1) = \frac{1}{(-1)^2} = 1 \)

Since \( f(1) = f(-1) = 1 \), we can see these distinct inputs produce the same output, making it a valid counter-example against the function being injective.

Distinct Inputs and Outputs

Distinct inputs resulting in distinct outputs is the hallmark of a one-to-one function. This distinctiveness ensures a clear and unique correspondence between elements of the function's domain and co-domain.

In simpler terms, for each output value, there must be precisely one input value that maps to it. If any two different inputs lead to the same output, the function loses its injective property.

• This is crucial for maintaining the uniqueness of inputs through transformations.
• In the case of the given function \( f(x)=\frac{1}{x^2} \), failures to maintain distinct outputs illustrates its non-injectivity.