Understanding the Domain and Range of a Quadratic Function: Case Study Y x^2

Quadratic functions, such as the one defined by the equation Y x^2, are a fundamental concept in mathematics with significant applications in various fields including physics, engineering, and data analysis. This article delves into the specific properties and characteristics of the function Y x^2, exploring its domain and range, and providing a comprehensive summary of its behavior.

Domain and Range of Y x^2

The function Y x^2 is a classic quadratic function characterized by its parabolic shape. Let's break down the domain and the range of this function in detail.

Domain

The domain of a function refers to the set of all possible input values (in this case, the range of values that x can take). For the function Y x^2, x can take on any real number value. Mathematically, the domain is expressed as:

Domain: (-∞, ∞)

Interval notation for the domain is (-∞, ∞). In simpler terms, this means that there are no restrictions on the values of x; it can be any real number.

Range

The range of a function refers to the set of all possible output (or y) values that the function can produce. For the function Y x^2, the key observation is that the output y is always non-negative, ranging from zero to infinity. This is because squaring any real number (positive, negative, or zero) results in a non-negative value.

Range: [0, ∞)

The range is thus expressed in interval notation as [0, ∞)/[0, ∞). This means that the smallest value of y is 0, and it can take any value greater than 0 and extend to infinity.

Summary

For the function Y x^2, the domain is all real numbers, while the range is all non-negative real numbers.

Comparative Analysis with Exponential Functions

Exponential functions, such as y 2^x, also have interesting properties regarding their domain and range. Let's explore these functions in a similar manner to the quadratic function:

Domain and Range of y 2^x

The exponential function y 2^x is defined for every real number x; thus, its domain is all real numbers.

Domain: (-∞, ∞) or R

Since the exponential function y 2^x grows without bound as x increases and approaches 0 as x decreases, the function can take any positive value. Therefore, the range of the exponential function is all positive real numbers:

Range: (0, ∞)

In summary, while the quadratic function Y x^2 can produce values from 0 to infinity, the exponential function y 2^x can produce values from 0 (exclusive) to infinity, reflecting the different characteristics of these two types of functions.

Application in Specific Domains

The function y x^2 with a restricted domain, such as -1 ≤ x ≤ 4, can be analyzed to provide specific examples of domain and range:

Restricted Domain: -1 ≤ x ≤ 4

For the given function y x^2 with the domain restricted to -1 ≤ x ≤ 4, the domain is expressed as:

Domain: [-1, 4]

Within this domain, we can compute the values of y at the endpoints:

When x -1, y (-1)^2 1 When x 4, y (4)^2 16

Thus, the range of the function is:

Range: [1, 16]

Using the Intermediate Value Theorem (IVT), we can conclude that the function attains all values between 1 and 16.

General Parabolic Function y x^2

The parabolic function y x^2 represents a parabola with its vertex at (0, 0) opening upwards. This function is defined for all real values of x, making its domain the entire set of real numbers. The output y is always non-negative, as x^2 is either zero or positive:

Domain: (-∞, ∞) or R

Range: [0, ∞)

This indicates that the function can produce any non-negative value, but never a negative one.

Conclusion

In conclusion, understanding the domain and range of a function, such as Y x^2, is crucial for comprehending its behavior and applications in various contexts. The domain of the quadratic function is all real numbers, while the range is all non-negative real numbers. This article also highlights the differences in domain and range between quadratic and exponential functions, providing insights into the unique characteristics of these functions.