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In precalculus, a function is a special type of relationship between two sets that assigns to each element of the first set, exactly one element of the second set. Think of it as a unique pairing where every input, which is a member of the set known as the domain, is associated with exactly one output, a member of the range.

A common way to express a function is by using a function notation like f(x), where f represents the function name, and x is the variable representing any element from the domain. For example, a simple function could be defined as f(x) = x + 2, which means for every value x you give to the function, the output will be x increased by two.

The function in the exercise is quite special because it is defined only for a set of discrete points, namely {3, 4, 7, 9}. These kinds of functions are useful in real-world scenarios where a function is only relevant or defined at specific instances, such as a timetable for a bus that only runs at certain hours of the day.

The domain of a function is the set of all possible input values (or x-values) that the function can accept without resulting in any undefined or nonsensical expressions. In simpler terms, it’s all about what you can put into the function. For instance, if the function involves a square root, the domain would be all real numbers greater than or equal to zero because you can't take the square root of negative numbers (in the realm of real numbers).

The range, on the other hand, is about what comes out of the function. It's the set of all possible output values (y-values) the function can provide based on the domain. For the function in our exercise, the domain is explicitly given as the set {3, 4, 7, 9}, and the range is the set {-1, 0, 3}. Understanding the concept of domain and range is crucial because it lets you grasp the extent of a function — what it can handle and what it can produce.

A piecewise function is a function that's defined by multiple sub-functions, each applying to a certain interval or set of inputs in the domain. It's like a mathematical chameleon, changing its formula depending on the input. The function in this exercise is a classic example of a piecewise function because it specifies different outputs for different inputs.

In our example,

• For the input value 3, the function gives an output of -1.
• If you input 4, you get an output of 0.
• And for input values 7 and 9, the function outputs 3.

This piecewise definition is necessary because there isn't one single rule that applies across the entire domain; each part of the domain has its own rule. Piecewise functions can model various real-life situations, like tax rates that change based on income brackets, or parking charges that vary by the time of day.

The representation via distinct cases, as in the provided exercise, is what makes the piecewise function a particularly versatile tool in mathematics. They allow for the explanation of complex, real-world phenomena with precision and clarity.