91Ó°ÊÓ

The absolute value function is a fundamental concept in mathematics, represented by the notation \( |x| \). It essentially measures the distance of a number from zero on the number line, regardless of direction.
For example, both \( 3 \) and \( -3 \) have an absolute value of \( 3 \). When dealing with \( y = |\sqrt{x} - 2| \), it means any negative result from \( \sqrt{x} - 2 \) will be flipped to a positive value by the absolute value function.
This reflection property is visually observed in graphs as it reflects parts of a curve that dip below the x-axis back above it. Thus, it ensures that all y-values are non-negative, which is essential in analyzing mixed functions like our given equation.

The square root function, represented as \( y = \sqrt{x} \), is a standard mathematical function that produces the principal square root of x.
It shows how numbers map to their square roots, which is always positive for non-negative x-values.

• The domain of the square root function is all non-negative numbers (\( x \geq 0 \)).
• The range is also non-negative, which means outputs are always \( y \geq 0 \).
• The graph of \( y = \sqrt{x} \) is a smooth curve starting at the origin \((0, 0)\).

In our exercise, the square root function provides the basis upon which other operations are performed. As part of the function \( |\sqrt{x} - 2| \), the square root operation applies first before considering its modification by subtracting 2 and taking the absolute value.

A piecewise function is a function that has different expressions based on the input value x. This type of function can allow a graph to change behavior at certain points.
In the expression \( y = |\sqrt{x} - 2| \), we can see the behavior changes depending on whether \( \sqrt{x} \) is less than or greater than 2.
The absolute value causes the function to behave differently in two "pieces":

• When \( \sqrt{x} < 2 \) (or \( x < 4 \)), the function becomes \( y = 2 - \sqrt{x} \).
• When \( \sqrt{x} \geq 2 \) (or \( x \geq 4 \)), the function is \( y = \sqrt{x} - 2 \).

Understanding the piecewise nature of the function helps to outline its complete graph, displaying its unique hook-like form with distinct sections that transition smoothly from one behavior to another at \( x = 4 \).

To effectively graph any function, identifying key points is crucial. This process involves calculating specific points that provide a framework for the graph.
For \( y = |\sqrt{x} - 2| \), we determined critical points by substituting particular values of x:

• \((0, 2)\): When \( x = 0 \), the value of y is \( 2 \).
• \((4, 0)\): Setting \( x = 4 \) gives y as \( 0 \).
• \((1, 1)\): With \( x = 1 \), y is \( 1 \).

These points guide the sketch of the function's graph. Starting at \( (0, 2) \), the graph descends towards \( (4, 0) \) and then ascends upwards again.
By plotting these and considering the function’s characteristics - like its piecewise nature - we create an accurate visual representation that helps in understanding the function's behavior.