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The **domain of a function** refers to all the possible input values that a function can accept. For example, if you think about feeding your pet only certain foods, the domain is like the list of acceptable foods you can give to your pet. In a mathematical function, the domain includes all the values of \( x \) for which the function is defined.

Let's look at function \( f \) from our exercise. Its domain consists of the numbers

• -1
• 0
• 1
• 2
• 3

These are the only inputs for which we can calculate \( f(x) \). Similarly, the domain of function \( g \) consists of

• 0
• 1
• 2
• 3
• 4

Understanding the domain is crucial because if you try to use a value outside of this set, like trying to find \( f(4) \), it would be like trying to feed your pet chocolate—it's just not allowed.

When composing functions, the domain of the composite function \( f \circ g \) is determined by the domain of \( g \) but further restricted by the domain of \( f \). This means \( (f \circ g)(x) \) is only defined if \( g(x) \) is a part of the domain of \( f \). This is important for ensuring that every step of function composition works smoothly.

The **range of a function** is the set of all possible outputs. Just like the domain tells us where we can start, the range tells us where we can end up. Think of it like the destinations you can reach with the vehicles available to you. With a function, this is the collection of values \( f(x) \) can take.

For instance, in the table for function \( f \), we have outputs:

• 2
• 0
• 3
• 1

This is the range of \( f \). Similarly, the range of function \( g \) includes

• 3
• 2
• 0
• 4
• -1

In the context of composing functions like \( f \circ g \), the range of the final composition also depends on the range of \( g \) and how those outputs fit into the domain of \( f \).

By understanding both functions’ domains and ranges, you'll see how the composite function only allows certain outputs, reflected in the composite function's range.

**Undefined values in functions** occur when an operation in a function doesn't result in a valid real number outcome. It's like trying to read a GPS map for a foreign city without an internet connection—you're missing essential details! In mathematics, something is undefined if it doesn't produce a meaningful result based on the given domain and operation conditions.

In our exercise, when we compute \( (f \circ g)(3) \), we see that \( g(3) = 4 \). However, when we try to find \( f(4) \), it becomes undefined because 4 isn't in the domain of \( f \). Similarly, for \( g \circ f \), \( f(4) \) is undefined, making \( (g \circ f)(4) \) undefined as well.

This highlights the importance of respecting a function's domain and range to avoid undefined values. If an input or output steps outside the function's boundaries, the result is undefined. Remember to always check these limitations to know when a function might fail to provide a valid result.