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When we talk about the domain of a function, we refer to the complete set of possible input values (usually denoted by \( x \)) for which the function is defined. For a function to be defined, it generally means that we can plug in a value and get a valid output without any math errors.
One of the primary reasons a function might not be defined at a certain point is division by zero. In our exercise, the function \( G(x) = \frac{3x + |x|}{x} \) is defined for all real numbers except one value where \( x \) causes the division by zero. This critical value is \( x = 0 \), because dividing by zero is undefined in mathematics.
Thus, the domain of the function \( G(x) \) is all real numbers except zero, which we can express as \( x \in \mathbb{R} \setminus \{0\} \). This means any real number except 0 can be an input.

Piecewise functions are fascinating because they are defined by multiple sub-functions, each applied to a certain interval of the main function's domain. This means a piecewise function takes different forms depending on the value of \( x \).
In our exercise, the function \( G(x) = \frac{3x + |x|}{x} \) behaves differently based on whether \( x \) is positive or negative due to the presence of the absolute value component, \( |x| \).

• For \( x \geq 0 \), the absolute value \( |x| \) simplifies to \( x \), making the function \( G(x) = 4 \).
• For \( x < 0 \), \( |x| \) simplifies to \(-x \), turning the function into \( G(x) = 2 \).

This results in different portions of the function defined over different intervals, which is a perfect example of a piecewise function. Such functions are quite useful in real-world scenarios where different conditions apply at different ranges.

The absolute value function is a very special type of function. It always outputs the non-negative value of any number you input. It's like a machine that takes a number and makes it positive.
For any real number \( x \), the absolute value \( |x| \) is defined as:

• \( x \), if \( x \geq 0 \)
• -\( x \), if \( x < 0 \)

In the given exercise, \( G(x) = \frac{3x + |x|}{x} \) uses the absolute value within a larger expression. This requires us to consider different cases: \( x \geq 0 \) and \( x < 0 \), because the behavior of the absolute value changes depending on the sign of \( x \). Understanding this property of the absolute value function allows us to break down complex functions into simpler components, aiding both analysis and computation.

Graphing functions is a powerful way to visualize how they behave across their domain. It involves plotting points on a coordinate plane to represent the function's output for each input.
For the function \( G(x) = \frac{3x + |x|}{x} \), graphing involves drawing separate horizontal lines due to its piecewise nature.

• For \( x < 0 \), where \( G(x) = 2 \), you draw a horizontal line at \( y = 2 \) for all negative \( x \).
• For \( x > 0 \), where \( G(x) = 4 \), a horizontal line is drawn at \( y = 4 \) for all positive \( x \).

It's crucial to note that at \( x = 0 \), the function is undefined, creating a gap or dot that is not filled in on the graph. This visual gap is essential because it tells us where the function does not provide an output. Graphing not only aids in understanding the piecewise nature of the function but also highlights the discontinuity at certain points.