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Understanding function values is like finding treasure in a multi-dimensional space. Let's think of a function as a machine that takes an input and gives you an output based on specific rules.

In the context of multivariable functions, the rules are more complex because our 'machine' takes several inputs at once. For instance, the function \(f(x, y, z) = \sqrt{x+y+z}\) asks for three values, corresponding to \(x\), \(y\), and \(z\). To find the function value, also known as the output, we follow the formula and carry out the operations with the given inputs.

To better understand, imagine you have a basket with apples (\(x\)), bananas (\(y\)), and oranges (\(z\)). The function tells us to add one of each fruit to find out how 'heavy' the basket is. Though we don’t measure weight here, we measure the 'output' by adding the values of each fruit and then taking the square root of the total.

Discovering the Root of the Matter

When we talk about radical expressions, we're diving into the world of roots—square roots, cube roots, and so forth. The square root symbol (\(\sqrt{}\)) is the most recognized radical. It's like asking, 'What number multiplied by itself gives me this other number?'

In our exercise, when we see \(\sqrt{9}\) or \(\sqrt{11}\), we're looking for numbers that, when squared, yield 9 or 11, respectively. While \(\sqrt{9}\) is a clean 3, because 3 times itself gives 9, \(\sqrt{11}\) is not as tidy. It's an irrational number which means it cannot be expressed as a simple fraction, and its decimal form is non-repeating and non-terminating.

Imagine you’re a detective in a vast city and each suspect is a variable holding different pieces of information. Substituting variables is like swapping suspects to see which one fits the story our function is trying to tell.

Substitution is a process of replacing the variables in a function with actual values to figure out what the function's output is. Think of it as completing a puzzle. You have shapes and spaces where each piece fits, and in mathematical terms, each space is a variable—\(x\), \(y\), or \(z\)—awaiting the perfect numerical piece. Once you fit the correct number into its place, the bigger picture, or the value of the function, is revealed.