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To understand the domain of a function, think of it as the set of all possible input values for a function. For any given function, the domain tells you all the values that you can substitute for the variable, usually represented by \(x\). In the exercise, the function provided is the constant function \(f(x) = 0\). Since this function does not have any restrictions, you can input any real number into it.
To summarize, for the constant function \(f(x) = 0\), the domain includes:

• All real numbers
• Not limited to any specific interval
• Can be expressed in interval notation as \((-\infty, \infty)\)

Understanding the domain is crucial as it defines the scope of the function and helps in graphing and solving equations.

Now, let's move on to the range of a function. The range represents all possible output values that the function can produce. For a constant function such as \(f(x) = 0\), the output is always the same, regardless of the input value. This makes the concept of range straightforward. For the given function, the output is always zero.
In other words, the range of \(f(x) = 0\) includes:

• The single value 0
• Represented as \(\text{\{0\}}\)

This simplification is important because it helps in determining how the function behaves across its domain. Understanding the range is crucial for graphing because it tells you what values on the y-axis the function can touch or cross.

Let's focus on the concept of a constant function. A constant function is a type of function where the output value is the same for any input value. In mathematical terms, it can be written as \(f(x) = c\), where \(c\) is a constant number. For example, in the exercise, the constant function given is \(f(x) = 0\). This means that no matter what value of \(x\) you input into the function, the output will always be zero.
Here’s what you need to keep in mind about constant functions:

• The graph of a constant function is always a horizontal line
• The y-coordinate is the value of the constant
• For \(f(x) = 0\), the line is on the x-axis

Graphing a constant function is simple because you only need to draw a horizontal line. This consistency makes constant functions easy to understand and identify. Whether you’re asked to graph the function or determine its domain and range, the steps are intuitive and straightforward.