91Ó°ÊÓ

Understanding interval notation is crucial when defining the domain of a function. Interval notation is a way of representing a range of values on the number line. For open intervals, where the endpoints are not included, we use parentheses: \((-a, b)\). In contrast, for closed intervals, where the endpoints are included, we use brackets: \([a, b]\).

Remember:

• An open interval does not include its endpoints.
• A closed interval includes its endpoints.

For example, if a function is defined for all values between -5 and 5, not including -5 and 5, it would be written as \((-5, 5)\).
In the case of the given function, since there are no restrictions on \( x \), the domain includes all real numbers, which is represented in interval notation as \( (-\rightarrow, +\rightarrow) \).

Understanding when a function becomes undefined is essential in determining its domain. A function becomes undefined at values of \( x \) that make the denominator zero (since division by zero is undefined).
For the function \( h(x) = \frac{18x}{x^2 + 100} \), we need to find the values of \( x \) that satisfy \( x^2 + 100 = 0 \).
Solving:
\[ x^2 + 100 = 0 \]
This results in a negative term, meaning it has no real solutions because \( x^2 \) can never be negative. Therefore, for this particular function, there are no values of \( x \) that make it undefined.
As a result, the function is defined for all real numbers.

Real numbers are the set of numbers that includes all the numbers on the number line. These include both rational numbers (like fractions and integers) and irrational numbers (like the square root of 2 or \( \pi \)).

In mathematical notation, real numbers are often represented as \( \mathbb{R} \).
For our function \( h(x) = \frac{18x}{x^2 + 100} \), it is important to understand that we're only looking at real values for \( x \). Since no real values make the denominator zero, every real number is valid for \( x \).

Therefore, in interval notation, the domain of this function is all real numbers, written as \((-\infty, +\infty)\).