Problem 19 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The function is not one-to-one because different inputs can yield the same output.

Step 1: Understand the definition of a one-to-one function

A function is one-to-one if each output value can be attributed to exactly one input value. This implies that no two different inputs yield the same output.

Step 2: Analyze the Function

Consider the function given, \(f(x) = \frac{1}{x^2}\). Analyze if different input values \(x_1\) and \(x_2\) can produce the same output value.

Step 3: Test Input Values

Test the function with different pairs of input values. For instance, let \(x_1 = 1\) and \(x_2 = -1\). Then, \(f(1) = \frac{1}{1^2} = 1\) and \(f(-1) = \frac{1}{(-1)^2} = 1\). Both inputs produce the same output, meaning the function is not one-to-one.

Step 4: Conclude from the Analysis

Since different input values \(x_1\) and \(x_2\) can result in the same output, the function \(f(x) = \frac{1}{x^2}\) is not one-to-one.

Key Concepts

Functions

Functions form the backbone of mathematics by relating a set of inputs to a set of possible outputs. Think of a function as a machine or a box. You put something into this box (the input), the machine does something with it, and out comes the result (the output). Each input value has at most one output value. This property makes functions predictable and manageable.
For example, with the function \( f(x) = x + 2 \), if you input \( x = 3 \), the output is \( 5 \). This is because you added 2 to the input value.
Functions are everywhere around us: in physics, economics, engineering, and much more. They help us describe relationships and predict future events or behaviors based on current or past inputs.

Mathematical Analysis

Mathematical analysis is an advanced and detailed study involving calculus and functions to solve complex mathematical problems. It involves understanding the behavior of mathematical functions and by extension, their outputs.
Analyzing functions requires checking properties like continuity, limits, and differentiability. One important aspect is determining whether a function is one-to-one. In mathematical terms, a function \( f(x) \) is one-to-one if different inputs always result in different outputs.
In the exercise, we submitted two different inputs, \( x_1 = 1 \) and \( x_2 = -1 \), into the function \( f(x) = \frac{1}{x^2} \) and observed that they produced the same output. This use of analysis confirmed that our function is not one-to-one.

Input and Output Values

In functions, each output is determined by an input. If you think of a function as a machine, the input values are what you provide to the machine and the output values are what the machine gives back.
This relationship is crucial, especially in determining the nature of functions. A function is one-to-one if for every unique input, there is a unique output.
In our specific exercise with the function \( f(x) = \frac{1}{x^2} \), we examined how different inputs such as \( x = 1 \) and \( x = -1 \) led to the same output of 1. This means the input values \( x_1 \) and \( x_2 \) aren't always directly distinguishable based on their respective outputs. As a result, the function isn't one-to-one, indicating that different paths lead to identical destinations.