91Ó°ÊÓ

The fifth root function is represented mathematically as \( \text{\raisebox{2pt}{\(\root 5\thickspace \box 0\)} f(x)} = \text{\raisebox{2pt}{\(\root 5\thickspace \box 0\)} (2t + 7)} \). It involves finding a number that, when raised to the power of 5, gives the original number inside the root. Unlike square roots, which only accept non-negative numbers, fifth roots can accept any real number. This is because any real number, negative or positive, raised to the fifth power remains a real number. Consequently, the fifth root function is particularly flexible in terms of the numbers it can process. For instance, in the function given, \( f(t) = \text{\raisebox{2pt}{\(\root 5\thickspace \box 0\)} (2t + 7)} \), there is no restriction on the value of \( t \).

Identifying the domain of a function involves determining all possible input values (commonly \( x \) or \( t \)) for which the function is defined. In simpler terms, it's about finding which values you can plug into the function without causing any mathematical issues like division by zero or taking the square root of a negative number. For the fifth root function \( \text{\raisebox{2pt}{\(\root 5\thickspace \box 0\)} (2t + 7)} \), there are no such problems. This is because any real number under the fifth root sign is valid. Therefore, the function \( f(t) = \text{\raisebox{2pt}{\(\root 5\thickspace \box 0\)} (2t + 7)} \) is defined for all real numbers. In mathematical notation, the domain of \( f(t) \) is \( (-\boldsymbol\textinfty, +\boldsymbol\textinfty) \).

Real numbers include all the numbers you can think of except for imaginary numbers. They encompass positive and negative integers, fractions, and irrational numbers. Basically, it’s any number that you can find on the number line. In dealing with functions, especially functions that involve roots, it is crucial to understand the behavior of real numbers under certain mathematical operations. For example:

• Positive Numbers: These are numbers greater than zero.
• Negative Numbers: These are numbers less than zero.
• Zero: This is also a real number and is neutral.
• Fractions and Decimals: These are also real numbers.

Understanding real numbers helps in determining the domain of various functions, such as the fifth root function in our exercise, which accepts all real numbers.