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In mathematics, the domain of a function is the set of all possible input values (or x-values) that allow the function to work properly without any errors. For the function given, \[G(x)=\frac{3x+|x|}{x}\]we need to look out for conditions where the function becomes undefined. A common issue arises when the denominator of a fraction is zero. In this case, when \(x = 0\), the function is undefined because division by zero isn't possible.Therefore, the domain of this function is all real numbers except zero. We can express the domain with a set notation as:

• \( \{ x \in \mathbb{R} \mid x eq 0 \} \)

This means every real number is allowed except for zero. Always check denominators in functions to determine their domains effectively.

The absolute value function, noted as \(|x|\), measures the distance of a number from zero on the number line. It always gives a non-negative result, regardless of whether the input is positive or negative.For our function, \[3x + |x|\]the absolute value influences the piecewise behavior:

• When \(x \geq 0\), the absolute value behaves normally, and \(|x| = x\).
• When \(x < 0\), the absolute value acts to negate the input, giving \(|x| = -x\).

This characteristic splits the function into different expressions based on the sign of \(x\), which simplifies to different constant values on different intervals:

• For \(x \geq 0\), \(G(x)\) simplifies to 4.
• For \(x < 0\), \(G(x)\) simplifies to 2.

When approaching functions, it's crucial to identify any points where the function is not defined. In our function,\[G(x)=\frac{3x+|x|}{x}\]the function is undefined at \(x = 0\) due to division by zero.These undefined points mark gaps or interruptions in the overall function. To address undefined points:

• Identify the factors in the denominator.
• Check when these factors equal zero.
• Exclude these values from the domain to avoid errors.

For purposes like sketching a graph, undefined points often result in open circles on the plot to show these gaps.

Graph discontinuity occurs when there are breaks or jumps in the graph of a function. For our piecewise function:

• For \(x > 0\), \(G(x)\) is a constant line at 4.
• For \(x < 0\), \(G(x)\) is a constant line at 2.

At \(x = 0\), the function is not defined, creating a discontinuity point, evident as an open circle when graphing.This discontinuity shows that the values do not connect smoothly at this point and further highlights the separation between the two expressions of the piecewise function. Recognizing discontinuities helps better understand the behavior of functions and is crucial when analyzing piecewise functions graphically.