91Ó°ÊÓ

A fifth root function involves taking the fifth root of a number or expression. In mathematical terms, for any number or expression, the fifth root can be expressed as \(\root{5}\frac{(expression)}{5}\).
In our example, the function given is \(\frac{\root{5}{t+1}}{10}\). The fifth root can handle both positive and negative inputs. This is different from square root functions which only allow non-negative inputs. The fifth root of any real number exists because the power of five does not introduce any restrictions on the domain. Understanding this allows us to easily determine that our function can accept any real number as its input.

The domain of a function refers to all possible values that we can input into the function. For some functions, certain operations or values can cause the function to become undefined. For example, having zero in the denominator or a negative value under an even root (like a square root).
For the given function, \(\frac{\root{5}{t+1}}{10}\), we noticed two things: The fifth root \(\root{5}{t+1}\) can take any real number, positive or negative, without causing issues. Also, since the denominator is a constant 10 and not dependent on the variable \(\text{t}\), it will never be zero.
Thus, there are no domain restrictions caused by the denominator or the fifth root, meaning any real number is a valid input.

Real numbers include all the positive numbers, negative numbers, and zero. This also encompasses rational numbers (like fractions and integers) and irrational numbers (like \(\text{Ï€}\) and the square root of 2).
When we talk about domain in terms of real numbers, we are considering every possible value on the number line. Since the function \(\frac{\root{5}{t+1}}{10}\) can accept any \(\text{t}\) without making the function undefined, it means our function is defined for all real numbers.
Therefore, the domain of the function \(\frac{\root{5}{t+1}}{10}\) is all real numbers, written in interval notation as \((-\text{∞}, \text{ +∞})\).