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Understanding what defines a function in algebra is crucial for grasping various mathematical concepts. A function is essentially a special type of relation where every input, commonly denoted as 'x', is related to exactly one output, often referred to as 'y'. It's like a machine where you put in a number, and a single, specific number comes out.

In our exercise, we aimed to determine if an equation sets the stage for 'y' to be dependent on 'x', meaning for every 'x' value that we plug into the equation, we expect to get one and only one 'y' value in return. If this criterion isn't met, what we have isn't a function. The preliminary steps in the exercise demonstrate how 'y' is isolated to establish this relationship, resulting in the formula: \(y = \frac{1}{x + 2}\).

When working out if an equation describes a function, it's important to consider the domain, which in this case is all real numbers except where the denominator is zero. Since there is only a single value for 'x' that we must exclude (-2), for all other 'x' values there's a corresponding single 'y' value, and this satisfies the definition of a function.

Isolating a variable is a systematic approach to solving for a specific unknown in an equation. This process is vital when you're trying to express one variable explicitly in terms of another—commonly when formulating a function. In our textbook exercise, we isolate 'y' by first factoring 'y' out of the equation, and then dividing both sides by the remaining binomial that includes 'x'.

The details provided in the solution steps illustrate how isolating 'y' helps us reconstruct the equation, transforming it into a form that distinctly shows 'y' as a function of 'x'. This manipulation of algebraic expressions is not just a useful skill in solving textbook exercises but is foundational to understanding and describing relationships in mathematics, science, and engineering.

What makes a mathematical relation a function is strictly defined by certain criteria. The key rule here is that for every input, there should be exactly one output. A simple way to test this is using a vertical line test on a graph. If a vertical line intersects the graph more than once, then the graph does not represent a function.

In the context of our exercise, the function criteria are explored through analytic methods rather than graphing. The critical question asked was whether 'y' has one and only one value for each 'x', except where the function is undefined. After isolating 'y', it was discerned that the equation met the function criteria for all 'x' except at 'x = -2', where the function is undefined due to a zero in the denominator. This understanding informs us that the equation fulfills the function criteria with the mentioned exception, a caveat that is crucial in mathematical analysis and applications.