91Ó°ÊÓ

The fourth root function is a special type of root function where the exponent is 1/4. For any function of the form \[ g(x) = \sqrt[4]{x} \] it means we are looking for the value that, when raised to the power of 4, gives us x. Simply put, the fourth root of a number is a value that when multiplied by itself four times results in the original number. For instance, the fourth root of 16 is 2 because \[ 2^4 = 16 \]. In the given exercise, the function is \[ g(t) = \sqrt[4]{t+8} \], which means we are dealing with the fourth root of the expression t+8. To ensure the root is defined for all values of t, the expression inside the root must be non-negative (0 or positive). This leads us to the next key topic: inequalities.

An inequality is a mathematical statement that explains the relative size or order of two values. Inequalities use symbols like > (greater than), < (less than), \geq (greater than or equal to), and \leq (less than or equal to). In the problem, to get a valid input for the fourth root function, we must ensure that the expression inside the root, t+8, is non-negative. Hence, we write the inequality: \[ t+8 \geq 0 \]. Solving this involves: 1. Subtracting 8 from both sides: \[ t \geq -8 \]. 2. This means that t must be greater than or equal to -8 for the function to have real values. Inequalities help us determine the range of acceptable values and are crucial in defining the domain of functions.

Interval notation is a way of writing subsets of the real number line. It’s a shorthand way to describe the set of all numbers between a start and an end point. There are three types of intervals: 1. Closed interval: Uses square brackets [ ], includes both endpoints. E.g., \[ -8, 5 \]. 2. Open interval: Uses parentheses ( ), excludes both endpoints. E.g., ( -8, 5 ). 3. Semi-open/Semi-closed intervals: Combines brackets and parentheses. E.g., [ -8, 5 ). In this exercise, we determined that t must be greater than or equal to -8. So, the domain in interval notation is \[ -8, \infty) \]. This notation tells us that t starts from -8 and goes to infinity, including -8 but not including infinity. Interval notation is essential for clearly communicating the domain of functions.