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Understanding the domain of a function is fundamental to graphing it accurately. For the function given, \(y = \sqrt{|x|}\), finding the domain involves identifying all possible input values of \(x\) that make the expression under the square root well-defined. The absolute value in \(\sqrt{|x|}\) ensures that whatever the value of \(x\), the expression within the square root is non-negative. This means the square root of a negative number, which isn't defined for real numbers, can never occur here. Hence, the domain of the function includes all real numbers; mathematically, this is expressed as \((-\infty, \infty)\). No matter the value of \(x\), positive or negative, \(y = \sqrt{|x|}\) will always yield a non-negative output.

When considering the range of the function \(y = \sqrt{|x|}\), we explore all possible output values \(y\) can take. Since the square root function and the absolute value ensure non-negative outputs, the range will naturally be from zero upwards. Specifically, the smallest value that \(y\) can be is zero, which occurs when \(x = 0\). Furthermore, as \(x\) becomes larger in magnitude, whether positive or negative, \(y\) continually increases. Therefore, the range of this function is expressed as \([0, \infty)\), capturing all non-negative real numbers.

Function symmetry is about identifying patterns in a graph that repeat around a line or point. For the function \(y = \sqrt{|x|}\), symmetry is present about the y-axis. This kind of symmetry is referred to as even symmetry. If a point \((x, y)\) exists on the function, then its mirror image about the y-axis, \((-x, y)\), must also exist. This means the function exhibits a reflective symmetry, making the left side a mirror image of the right. As a result, visually, the graph resembles a 'V' shape on its side, showing balance around the vertical axis.

In terms of increasing and decreasing intervals, the graph of \(y = \sqrt{|x|}\) shows interesting behavior. Both arms of the 'V' shaped graph originate from the point \(x = 0\) and move upwards as you proceed along the x-axis, both to the left and right. For \(x \geq 0\), the function \(y = \sqrt{x}\) is clearly increasing, meaning that as \(x\) increases, so does \(y\). When considering negative \(x\), the graph considers \(y = \sqrt{-x}\), which still increases because \(-x\) is positive. Therefore, across the entire domain of \((-\infty, \infty)\), the function \(y = \sqrt{|x|}\) is never decreasing; it is always increasing.