91Ó°ÊÓ

The function provided in the exercise is a square root function within a fraction. When we talk about the square root function, we refer to the mathematical operation that finds a number which, when multiplied by itself, yields the original number.This operation causes certain constraints on the values of the input variable, since the input to the square root function must be non-negative.In general, the square root of a negative number is not considered a real number. Therefore, for the function to be real-valued, we need to ensure that the input inside the square root is greater than or equal to zero. In our function, since the square root is in the denominator, we need this input to strictly be greater than zero to avoid any undefined behavior.

Interval notation is a way of writing subsets of the real numbers and is commonly used to describe the domain and range of functions.In interval notation, we enclose numbers within brackets to indicate the start and end points of an interval:

• Round brackets, ( and ), are used to indicate that the endpoints are not included in the interval.
• Square brackets, [ and ], are used when the endpoints are included.

For our specific function, we determined that t must be greater than zero because it is under a square root in the denominator. Thus, the domain is expressed in interval notation as (0, ∞). This means t can take any positive value.

Functions with denominators impose specific restrictions that must be analyzed carefully to avoid undefined behavior. For our given function, we have: \[ f(t) = \frac{1}{\sqrt{t}} \]Here, the denominator is the square root of t. We need to ensure that this denominator is never zero because division by zero is undefined and causes the function to break down. It is important to remember:

• The square root function needs its argument to be non-negative (i.e., greater than or equal to zero).
• Since it's in the denominator, we must strictly use values greater than zero to keep the function defined and avoid division by zero.

Combining these restrictions, we find that the input value t must be in the range (0, ∞) to ensure the function is properly defined.