91Ó°ÊÓ

Understanding how to solve inequalities is crucial when you're trying to find the domain of a function. An inequality is much like an equation but instead shows that one value is larger or smaller than another, not necessarily equal. For instance, in the given exercise, we encounter the inequality \(x - 10 > 0\). To solve it, you treat it similarly to an equation, performing the same operations on both sides to isolate the variable \(x\).

However, there's an important thing to remember: if you ever multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. With the inequality \(x - 10 > 0\), we simply add 10 to each side to find that \(x > 10\). This gives us a crucial part of our domain, indicating that \(x\) must be a number greater than 10 in order for the function to be defined.

A radical function contains a variable within a root, such as a square root or cube root. The function in our exercise, \(f(x) = \frac{x+2}{\sqrt{x-10}}\), includes a square root in the denominator. When dealing with radical functions, especially square roots, we have to ensure that the value under the root is non-negative, because the square root of a negative number is not defined within the real number system.

For the domain of radical functions, you'll usually set the radicand—that's the expression under the root—greater than or equal to zero and solve for \(x\). In this case, however, because the radical is in the denominator, the radicand must be strictly greater than zero (\(x-10 > 0\)) to avoid division by zero. Remember, you cannot take the square root of zero in a denominator because it would make the entire function undefined.

The domain of a function includes all the input values (\(x\text{-values}\) that will produce a valid output. When finding the domain, certain restrictions may apply that limit which \(x\text{-values}\) you can use. In identifying domain restrictions, we have to consider denominators and radicals separately.

Firstly, denominators cannot be zero because division by zero is undefined. This is why we must solve the inequality \(x - 10 > 0\), to make sure that we never have zero in the denominator. Secondly, for radical functions, as mentioned before, the radicand (what’s under the radical sign) must be non-negative if the radical is an even index, like a square root, and must be positive if it's in a denominator.

Combining these restrictions gives us the domain of the function. In regards to our exercise, the domain for \(f(x)\) is all values of \(x\) that are greater than 10 to satisfy both the radical and the denominator, leading us to a domain of \(x > 10\). Domain restrictions ensure we only work with inputs that make the function both real and defined.