91Ó°ÊÓ

Extended interval notation is a way to describe a set of real numbers in a concise form. It uses brackets and parentheses to indicate which numbers are included in a set.

Understanding the symbols is key:

• Parentheses ( and ) are used to denote that endpoints are not included. For example, extbf{(a, b)} means all numbers between a and b, excluding a and b.
• Brackets [ and ] are used to denote that endpoints are included, such as extbf{[a, b]} which includes a, b, and all numbers in between.
• Infinity symbols, extbf{-∞} and extbf{∞}, are always used with parentheses because infinity is not a number that can be reached.

When we say a function is defined on extbf{(-∞, ∞)}, it means the function accepts all real numbers as inputs. This was the case for our original function since the denominator never reaches zero. Remember, when dealing with more complex functions, consider points where the function can become undefined when finding the domain.

Trigonometric functions are fundamental to understanding various mathematical and practical applications. The function used in our problem, extbf{sin(x)}, is just one instance. Here’s a quick overview of key trigonometric functions and their ranges:

• extbf{sin(x)} varies from [-1, 1] and oscillates with a period of 2Ï€.
• extbf{cos(x)} also varies from [-1, 1], following a similar periodic pattern.
• extbf{tan(x)} has a range of all real numbers and repeats every Ï€, but becomes undefined where cos(x) = 0.

When forming or examining functions with trigonometric components, always check which values might cause the denominator to become zero, leading to undefined function points. In this exercise, we concluded that the issue of an undefined cos(x) at -2 isn't possible within its range, simplifying the solution greatly.

Denominator conditions are crucial when determining the domain of a function, especially those involving fractions. A function is undefined wherever its denominator equals zero. Therefore, identifying these conditions allows us to outline where a function is valid.

For our given function \( f(x) = \frac{\sin(x)}{2 + \cos(x)} \), the goal was to ensure the denominator \( 2 + \cos(x) \) never equals zero.

Steps for analyzing denominator-related domain issues include:

• Set the denominator equal to zero and solve for x.
• Check if solutions lie within the permissible numeric range of any trigonometric elements involved.
• Exclude these solutions from the domain if necessary.

In our case, the equation \( 2 + \cos(x) = 0 \) simplifies to \( \cos(x) = -2 \). Knowing \( \cos(x) \) is limited to [-1, 1], this case confirmed no real values for x will make the denominator equal zero, meaning the function is defined across all real numbers \((-\infty, \infty)\).