Question 3: Find the Domain and Range of the Function

Understanding Domain: The domain of a function is the set of all possible input values. To find the domain of the function, consider it defined for all possible values of independent variables.

Example: Consider the function \( f(x) = x^3 \). The domain of \( f(x) = x^3 \) is all real numbers. The range of the function is defined by all the outputs of the function, obtained after substituting the domain values into the function.

Question: Find the domain and range of \( \sin(x) \).

Solution: The domain of \( \sin(x) \) is all real numbers. To find the range of \( \sin(x) \), we use the identity \(\text{We know that }\)\(sin{^2x}+cos{^2x}=1\) \(\text{from the identity, we get}\) \(cos{^2x}=1-sin{^2x}\)\[\] \(cos{x} = \sqrt{1-sin{^2x}}\)\[\] We know that the cosine function is defined only for real values. therefore the value inside the root is always non-negative. \({1-sin{^2x}}\geq 0\) \(\text{implies that}\) \(\sin{x}\in(-1,1)\)\(\text{this is the Range of the } sin{x}\text{ function}\)