Problem 11 - Thomas Calculus

Question

Graph the functions in Exercises \(7-18 .\) What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. $$ y=\sqrt{|x|} $$

Step-by-Step Solution

Answer

The graph of \(y = \sqrt{|x|}\) is symmetrical about the y-axis, and the function is always increasing over its entire domain.

Step 1: Understand the Function

The function given is \(y = \sqrt{|x|}\). This is the square root of the absolute value of \(x\). The absolute value ensures that the input to the square root function is non-negative, which is necessary because the square root of a negative number would be undefined in the real number system.

Step 2: Identify Domain and Range

The domain of \(y = \sqrt{|x|}\) is all real numbers \(x\) because the absolute value of \(x\) ensures we always have a non-negative number inside the square root. Therefore, the domain is \((-\infty, \infty)\). The range of this function is \([0, \infty)\) because the square root outputs only non-negative numbers.

Step 3: Identify Symmetries

This function is symmetrical about the y-axis. For any point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\), which means the function exhibits even symmetry.

Step 4: Analyze Increasing and Decreasing Intervals

For \(x \geq 0\), \(y = \sqrt{x}\) which is an increasing function. For negative \(x\), \(y = \sqrt{|x|} = \sqrt{-x}\), which is also an increasing function because \(-x\) becomes positive as \(x\) becomes more negative. Thus, the function is always increasing over \((-\infty, \infty)\).

Step 5: Graph the Function

The graph of \(y = \sqrt{|x|}\) looks like a 'V' lying on its side, opening upwards. The left part of the 'V' is the graph of \(y = \sqrt{-x}\) for \(x < 0\), and the right part is \(y = \sqrt{x}\) for \(x \geq 0\). The graph is smooth and continuous, reflecting across the y-axis.

Key Concepts

Function Domain

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.

Function Range

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

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.

Increasing and Decreasing Intervals

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.