Problem 41 - Thomas Calculus

Question

Graph the functions in Exercises \(37-46 .\) 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 function is symmetric about the y-axis, increasing on \([0, +\infty)\), and decreasing on \((- fty, 0]\).

Step 1: Identify the Function

We are given the function \( y = \sqrt{|x|} \). This function involves both a square root and an absolute value. The absolute value makes it defined for all real numbers, but the square root restricts it to non-negative outputs.

Step 2: Analyze the Domain

The domain of \( y = \sqrt{|x|} \) is all real numbers \( x \), as the absolute value ensures that what's inside the square root is non-negative. Therefore, \( x \) can take any real number value.

Step 3: Identify Symmetries

The function \( y = \sqrt{|x|} \) is symmetric with respect to the y-axis. This is because replacing \( x \) with \(-x \) results in the same output: \( y = \sqrt{|-x|} = \sqrt{|x|} \). Therefore, the graph is even.

Step 4: Determine Intervals of Increase and Decrease

Since \( y = \sqrt{|x|} \) consists of two parts: \( y=\sqrt{x} \) for \( x \geq 0 \) and \( y=\sqrt{-x} \) for \( x < 0 \), we analyze each section separately. For \( x \geq 0 \), the function is increasing since roots of larger numbers are larger. For \( x < 0 \), the function is decreasing as the roots of smaller \(|x|\) values are smaller.

Step 5: Conclusion on Intervals

The function is increasing on the interval \([0, \infty)\) and decreasing on the interval \((-fty, 0]\). This is due to the behavior of the square root function combined with the absolute value.

Key Concepts

Symmetry in Functions

Symmetry in functions is a useful property that helps us understand the shape and behavior of a graph. For the function \( y = \sqrt{|x|} \), symmetry is evident. This function is symmetric with respect to the y-axis. What does this mean? If a function is symmetrical about the y-axis, replacing \( x \) with \(-x\) yields the same output value. Therefore, the graph looks identical on both sides of the y-axis. This type of symmetry is called even symmetry.

Checking for y-axis symmetry is simple. You can substitute \(-x\) for \(x\) in the function: \( y = \sqrt{|-x|} \), which simplifies back to \( y = \sqrt{|x|} \). Since there is no change, the graph is confirmed as even.

• The graph looks the same on either side of the origin.
• The function is unaffected by changing the sign of \(x\).

Domain and Range Analysis

Domain and range are fundamental concepts in understanding functions. For the function \( y = \sqrt{|x|} \), the domain includes all real numbers because the absolute value inside the square root ensures that the expression is always non-negative. Thus, you can plug in any real number for \( x \), and you will obtain a valid output.

The range, however, is limited to non-negative values. This is because a square root cannot yield a negative result. As such, the function will only produce outputs \( y \geq 0 \). So:

• Domain: All real numbers \( x \in (-\infty, \infty) \).
• Range: Non-negative real numbers \( y \in [0, \infty) \).

Intervals of Increase and Decrease

Understanding where a function is increasing or decreasing helps visualize the graph's shape over different intervals. For the function \( y = \sqrt{|x|} \), we need to look at its behavior on each side of the y-axis.

To the right of the y-axis (\( x \geq 0 \)), the function simplifies to \( y = \sqrt{x} \). Here, as \( x \) increases, \( y \) also increases, making the function strictly increasing on this interval.

On the left side (\( x < 0 \)), the expression becomes \( y = \sqrt{-x} \). In this situation, as \( x \) approaches zero from the left, the magnitude of \( x \) decreases, causing \( y \) to increase. Thus, the function decreases as we move to the left.

• Increasing Interval: \([0, \infty)\)
• Decreasing Interval: \((-\infty, 0]\)

This analysis gives a complete picture of how the function behaves over its entire domain.