91Ó°ÊÓ

Understanding the radicand in functions is essential when determining a function's domain, especially for functions involving roots, such as square roots or cube roots. In mathematical expressions, the radicand is the number or expression inside the radical symbol (for example, in \( \sqrt{x+3} \) the radicand is \( x+3 \) ). To find the domain of functions containing a radicand, we need to ensure that the value under the radical is appropriate for the type of root being taken. For square roots and even-indexed roots, the radicand must be non-negative to produce real numbers. However, for cube roots and other odd-indexed roots, we can have negative values as well.

When it comes to our exercise, we are dealing with the function \( f(x) = (x+3)^{3/2} \) where the radicand is \( x+3 \) and is under a cube root (implied by the denominator 3 in the exponent) followed by a square (implied by the numerator 2 in the exponent). This requires us to ensure the radicand is non-negative for the square root component, which dictates the domain of the function.

Inequalities are a fundamental component of understanding mathematical concepts such as the domain of a function. An inequality compares two values, indicating whether one is greater than, less than, or equal to the other. Solving inequalities is a process similar to solving equalities (equations), but with the additional consideration that multiplying or dividing both sides by a negative number reverses the inequality sign.

The core principle in dealing with inequalities in the realm of functions is to ensure that the function's expression is defined for all values within a certain interval. For our example, the inequality \( x + 3 \geq 0 \) derives from the necessity of keeping the radicand non-negative. Solving this inequality, we subtract 3 from both sides to isolate x, which gives us \( x \geq -3 \)—a statement describing all values of x that satisfy the condition we need for the function to be real-valued. Recognizing and solving these inequalities is critical to successfully identifying the domains of all kinds of functions.

Interval notation is a succinct way of representing the domain or range of a function, which includes sets of numbers. This notation uses brackets and parentheses to indicate closed or open intervals. A closed interval, denoted by square brackets [ ], includes its endpoints, whereas an open interval, denoted by parentheses ( ), does not include its endpoints.

In our exercise, the domain is expressed as \( [ -3, \infty ) \). The square bracket at -3 indicates that -3 is included in the set, marking it as the lower bound of the domain. Conversely, the parenthesis at \( \infty \) tells us that the domain extends indefinitely in the positive direction but does not include infinity itself since infinity is not a specific value but rather a concept describing unboundedness. Understanding interval notation is key to reading and writing the domain and range of functions clearly and correctly.