Understanding Domain and Range of Functions: Analyzing Specific Examples

The domain and range of a function are fundamental concepts in mathematics that describe the set of input values (domain) for which a function produces real outputs (range). In this article, we will explore two specific examples to thoroughly understand how to determine the domain and range of functions:

Example 1: y sqrt{4 - x^2}

Let us analyze the function y sqrt{4 - x^2}.

1. Finding the Domain

The expression inside the square root, 4 - x^2, must be greater than or equal to zero for the function to be defined:

4 - x^2 > 0

By solving this inequality:

4 > x^2

Taking the square root of both sides:

-2

Thus, the domain of the function is:

Domain: [-2, 2]

2. Determining the Range

The range can be found by determining the possible values that y can take:

When x -2 or x 2:

y sqrt{4 - (-2)^2} sqrt{0} 0

When x 0:

y sqrt{4 - 0^2} sqrt{4} 2

Since 4 - x^2 represents the upper half of a circle with a radius of 2 centered at the origin, y can take any value from 0 to 2.

Therefore, the range of the function is:

Range: [0, 2]

Example 2: y sqrt{-4x}

In the set of real numbers, the square root of a negative number does not exist. Therefore, for x to be defined, we have:

-4x > 0

Simplifying this inequality:

-x > 0

x

Thus, the domain of the function is:

D {x in mathbb{R} | x

Since the range uses the non-negative square root, we have:

R {y in mathbb{R} | y > 0}

General Approach to Domain and Range

To determine the domain and range of a function, follow these steps:

Analyze the function for any restrictions, such as square roots or denominators that cannot be zero. Identify the set of all possible input values (domain) that make the function valid. Determine the set of all possible output values (range) by analyzing the function for its minimum and maximum values. Use graphing techniques or calculus to find critical points and analyze the behavior of the function.

Conclusion

Understanding the domain and range of a function is crucial for comprehending its behavior and limitations. By following a structured approach, we can accurately determine these values for complex and simple functions alike.