Problem 76 - Thomas Calculus

Question

Graph the function \(y=\sqrt{|x|}\)

Step-by-Step Solution

Answer

The graph of \(y=\sqrt{|x|}\) resembles a 'V' shape with both arms being square root curves extending from the origin.

Step 1: Understand the Function

The function given is \(y = \sqrt{|x|}\). This involves both absolute value and square root operations. The expression inside the square root is \(|x|\), which means it is always non-negative. Hence, the domain includes all real numbers.

Step 2: Consider the Absolute Value Component

The absolute value \(|x|\) dictates that whether \(x\) is positive or negative, the value inside the square root will always be positive or zero. Thus, mathematically it can be split into two cases: \(x\geq0\) and \(x<0\).

Step 3: Examine the Function for \(x \ge 0\)

For \(x \geq 0\), \(|x| = x\) and so the function simplifies to \(y = \sqrt{x}\), which is the standard square root function for non-negative \(x\).

Step 4: Examine the Function for \(x < 0\)

For \(x < 0\), \(|x| = -x\) and so the function becomes \(y = \sqrt{-x}\). This part is similar to \(y = \sqrt{x}\) but mirrors it across the y-axis since the domain remains the same.

Step 5: Plot Key Points

Let's choose key points for plotting. For instance, when \(x = 0\), \(y = 0\). For \(x = 1\) or \(x = -1\), \(y = 1\). For \(x = 4\) or \(x = -4\), \(y = 2\).

Step 6: Sketch the Graph

Draw the graph based on the points and behavior identified. For \(x \geq 0\), plot the curve similar to \(y = \sqrt{x}\) in the first quadrant. For \(x < 0\), reflect this curve over the y-axis. The graph should form a 'V' shape.

Key Concepts

Absolute Value Function

The absolute value function, denoted as \(|x|\), is a fundamental concept in mathematics. It represents the distance of a number from zero on the number line, regardless of direction. This makes it always non-negative, which is crucial when combined with other functions.

• For positive numbers, \(|x| = x\).
• For negative numbers, \(|x| = -x\).

Understanding this concept helps in analyzing expressions involving absolute values because they do not alter positive inputs and change negative inputs to their positive counterparts. In function graphing, the absolute value modifies the input before any other operations, such as taking a square root. This is why expressions like \(|x|\) remain non-negative, thus directly influencing the domain of other functions that interact with it, such as the square root function. It is this property that keeps the domain of \(|x|\) open to all real numbers, resulting in particular symmetry properties in graphs, like the 'V' shape seen with \(|x|\) in certain contexts.

Square Root Function

The square root function, expressed as \(y = \sqrt{x}\), is another fundamental function. It is defined only for non-negative real numbers because you cannot take the square root of a negative number in the realm of real numbers. The output of a square root is always non-negative.

• The domain of \(y = \sqrt{x}\) is \([0, \infty)\).
• The range is also \([0, \infty)\).
• The function is increasing and represents half of a parabola rotated by 90 degrees.

In graphing this function, the points start from the origin \((0,0)\) and curve upwards to the right. For the original exercise \(y = \sqrt{|x|}\), the square root operation happens after taking the absolute value of \(x\). Since \(|x|\) guarantees a non-negative input, the function is valid for all real \(x\), leading to its mirrored appearance across the y-axis in the graph. This results in a symmetric curve resembling a sideways 'V'.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input values. These functions apply different rules to different parts of their domain, dividing the number line into sections where each section follows a unique expression. They are particularly useful when modeling situations that have abrupt changes.

• Each piece of the function is valid for a particular interval or condition.
• The boundaries between intervals are key to understanding the function's behavior.
• An example includes cases such as \(f(x) = x^2\) for \(x \geq 0\) and \(f(x) = -x^2\) for \(x < 0\).

In terms of graphing, piecewise functions can present multiple graphs within the same coordinate plane, highlighting different behavior for different intervals of \(x\). In the function \(y = \sqrt{|x|}\), although not explicitly a piecewise function, it inherently acts like one by behaving differently based on whether \(x\) is positive or negative. Understanding these divisions and behaviors helps in accurately sketching and interpreting the graphs of such functions.