Find the Domain and Range sin(x)^2

Solution:

Step 1: Identifying the Domain

The domain consists of all values that x can take such that the function is defined. For ${\sin }^2(x)$, there are no restrictions on x since the sine function accepts all real numbers and squaring it does not introduce any undefined points. Therefore, the domain is all real numbers.

Interval Notation: $(-\mathrm{\infty },\mathrm{\infty })$ Set-Builder Notation: $\{x|x\in \mathbb{R}\}$

Step 2: Determining the Range

To find the range, we look at the possible values of $y={\sin }^2(x)$. Since the sine function oscillates between -1 and 1, squaring it will result in values ranging from 0 to 1. Thus, the range includes all numbers from 0 to 1, inclusive.

Interval Notation: $[0,1]$ Set-Builder Notation: $\{y|0\le y\le 1\}$

Step 3: Combining Domain and Range

After evaluating both the domain and range, we can state them together.

Domain: $(-\mathrm{\infty },\mathrm{\infty })$, $\{x|x\in \mathbb{R}\}$ Range: $[0,1]$, $\{y|0\le y\le 1\}$

Knowledge Notes:

The domain of a function refers to the complete set of possible values of the independent variable (usually x) for which the function is defined. In the case of ${\sin }^2(x)$, since the sine function is defined for all real numbers and squaring the result does not introduce any additional restrictions, the domain is all real numbers, which is represented in interval notation as $(-\mathrm{\infty },\mathrm{\infty })$ and in set-builder notation as $\{x|x\in \mathbb{R}\}$.

The range of a function is the complete set of all possible resulting values of the dependent variable (usually y), after we have substituted the domain. In the case of ${\sin }^2(x)$, since the sine function has a maximum value of 1 and a minimum value of -1, when squared, the output range is from 0 to 1. This is because squaring any real number results in a non-negative number, and since the sine function cannot exceed 1 in magnitude, the squared value cannot exceed 1. The range is thus given in interval notation as $[0,1]$ and in set-builder notation as $\{y|0\le y\le 1\}$.

Interval notation is a way of writing subsets of the real number line. An interval notation consists of a pair of numbers that define the endpoints of the interval, and can be inclusive (using square brackets) or exclusive (using parentheses) of the endpoints.

Set-builder notation is a mathematical notation for describing a set by stating the properties that its members must satisfy. For example, $\{x|x\in \mathbb{R}\}$ reads as "the set of all x such that x is a real number."

It is important to understand the behavior of trigonometric functions, such as the sine function, and their transformations, such as squaring, to determine the domain and range of more complex functions.