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The square root function is a fundamental mathematical concept where the function is defined as \( y = \sqrt{x} \). This function provides the non-negative value that, when multiplied by itself, equals \( x \). In simpler terms, it is finding the number that when squared gives the original number.
This function is characterized by its increasing pattern as \( x \) grows larger, beginning with a flat slope and gaining steepness.

• The domain of a simple square root function \( y = \sqrt{x} \) is non-negative numbers \( x \geq 0 \), since a square root of a negative number is not defined in the set of real numbers.
• The range is also \( y \geq 0 \), as square roots yield non-negative results.
  
• The shape of a square root graph is often described as half of a sideways parabola.

The presence of the absolute value in the function \( y = \sqrt{|x|} \) alters this behavior slightly by allowing input values to include negative numbers.

The absolute value of a number \( x \) is denoted as \( |x| \) and represents the distance of that number from zero on the number line. It essentially strips away any negative sign, making every result non-negative.
This is crucial when interpreting functions involving absolute values, as it changes the behavior of the function across different x-values.

• For positive values of \( x \), \( |x| = x \).
• For negative values of \( x \), \( |x| = -x \).
• For zero, \( |x| = 0 \).

When used within the function \( y = \sqrt{|x|} \), the absolute value ensures that all inputs, regardless of whether they were originally negative, become non-negative before the square root is taken. This property ensures the graph is symmetrical about the y-axis, because \( |x| \) treats both positive and negative values in the same manner.

The domain of a function is the complete set of possible values of the independent variable, often denoted as \( x \), for which the function is defined. For different types of functions, the domain can vary:

• Polynomial functions have all real numbers as their domain.
• Rational functions have restrictions where the denominator approaches zero.
• Logarithmic and square root functions have their domain restricted to positive numbers because they do not exist in real numbers for negative inputs.
  

In the case of the function \( y = \sqrt{|x|} \), the absolute value ensures that \( |x| \) outputs non-negative numbers, thereby making the domain all real numbers: \( x \in (-\infty, \infty) \). This is contrasting other square root functions that traditionally have a more limited domain.

Symmetry in graphing refers to a graph's mirror-like balance on either side of a specific line, often the y-axis or the origin. This property helps in understanding the behavior of graphs and predicting values:

• Y-axis symmetry occurs when replacing \( x \) with \( -x \) yields the same value for \( y \). It appears as a reflection of the graph across the y-axis.
• Origin symmetry exists when both introduced negatives, \( -x \), and \(-y \) produce the same graph, appearing as a rotation around the origin.
  

In the function \( y = \sqrt{|x|} \), symmetry occurs because of the absolute value \( |x| \), indicating that both positive and negative values of \( x \) give the same \( y \). Thus, the graph is symmetrical about the y-axis, reflecting identical behavior to the right and left from the y-axis, resulting in a V-shaped graph.