Problem 23 - Calculus: One and Several Variables

Question

Give the domain and range of the function. $$f(x)=\frac{1}{x^{2}}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\frac{1}{x^{2}}\) is all real numbers except 0 (\(-\infty < x < 0\) or \(0 < x < \infty\)) and the range of this function is all positive real numbers (\(0 < f(x) < \infty\)).

Step 1: Identify the Domain

First, we need to find all the possible values that \(x\) can take. Given the function is \(f(x)=\frac{1}{x^{2}}\), it's clear that \(x\) can be any real number except for 0, since division by zero is undefined. Therefore, the domain is all real numbers except 0, which can be expressed as \(-\infty < x < 0\) or \(0 < x < \infty\).

Step 2: Identify the Range

Next, we need to find all possible values that \(f(x)\) can take. Given that \(x^{2}\) is always positive or 0, but in this function \(x\neq 0\), hence the denominator can't be zero and is always positive, and since we are taking reciprocal, the reciprocal of any positive number is also positive. Therefore, the range of function \(f(x)\) is all positive real numbers or \(0 < f(x) < \infty\).

Key Concepts

Function Analysis

When analyzing a function, we look at its behavior and characteristics. This includes examining the domain, range, and other important features. Understanding these aspects helps us predict how the function behaves in different scenarios.

• Domain: The set of all input values (x-values) for which the function is defined.
• Range: The set of all output values (y-values) the function can produce.

Analyzing functions allows us to understand the limitations and possibilities of the function. For example, with the function \(f(x)=\frac{1}{x^{2}}\), pinpointing the domain and range gives us insight into what values \(x\) and \(f(x)\) can take.

Real Numbers

Real numbers encompass all numbers that are either rational or irrational, meaning nearly every number you can think of. They include:

• Integers: Whole numbers and their negatives (e.g., -2, 0, 3).
• Fractions: Parts of a whole expressed as a fraction (e.g., 1/2, -3/4).
• Irrational numbers: Numbers that cannot be expressed as a fraction (e.g., \(\pi\), \(\sqrt{2}\))

The function \(f(x)=\frac{1}{x^{2}}\) is defined for almost all real numbers, except zero, because division by zero is not possible. Understanding real numbers helps to define where our function "lives" or which inputs it accepts.

Reciprocal Function

A reciprocal function involves flipping the position of numbers in a fraction. For any non-zero real number \(a\), its reciprocal is \(\frac{1}{a}\). This function is often represented in the form \(f(x)=\frac{1}{x}\) or with variations like \(f(x)=\frac{1}{x^{2}}\), as seen in our problem.
The key features of a reciprocal function are:

• Always defined except for zero: The function cannot take zero as an input since the reciprocal involves division.
• Produces positive outputs for positive inputs: In \(f(x)=\frac{1}{x^{2}}\), since \(x^2\) is always positive, \(f(x)\) is always positive.

By understanding reciprocal functions, we can better predict and navigate their unique behaviors and constraints.

Division by Zero

Division by zero is a crucial concept in mathematics, representing an operation that is undefined. If you attempt to divide by zero, the result is not a real number and calculations become impossible.
In the context of a function such as \(f(x)=\frac{1}{x^{2}}\), it's important to notice that \(x^2\) becomes zero when \(x=0\). This makes the expression undefined at \(x=0\).
Why is division by zero impossible?

• There’s no number that, when multiplied by zero, will give a non-zero number.
• It’s like trying to distribute a finite amount evenly in zero containers—it doesn’t make sense.

A full understanding of why division by zero is impossible helps in determining domains correctly in functions.