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When we discuss function continuity, we're looking at how smooth a function's graph is without any abrupt breaks or jumps. In this case, the function given is a piecewise function of the form \( y = |\sqrt{x} - 2| \). This function employs three operations: taking a square root, subtracting a number, and taking the absolute value.

Continuity depends on each of these operations:

• The square root function, \( \sqrt{x} \), is continuous for non-negative values of \( x \). This means no gaps or jumps for \( x \geq 0 \).
• Subtracting 2 from \( \sqrt{x} \) shifts the graph downward by 2 units, but doesn't affect continuity.
• The absolute value function, \( |u| \), where \( u = \sqrt{x} - 2 \), reflects any negative values of the inside operation over the x-axis, but also does not cause discontinuities.

Thus, the function \( y = |\sqrt{x} - 2| \) remains continuous for all values within its domain \( x \geq 0 \). You can sketch its graph without lifting your pencil off the paper.

The domain of a function is all the possible values that the independent variable, usually \( x \), can take for the function to output real numbers. The function \( y = |\sqrt{x} - 2| \) depends heavily on the square root operation.

For \( \sqrt{x} \), \( x \) must be non-negative, meaning \( x \geq 0 \). Square roots of negative numbers do not produce real results, therefore any negative values are excluded from our domain.

This leaves us with a domain expressed as:

• \( x \geq 0 \)

Understanding the domain is crucial before performing any operations or sketching the graph. Essentially, it tells us where to start plotting and analyzing our function.

The intercepts of a graph are points where the graph crosses the axes. Identifying intercepts provides quick insights into the behavior of the function.

**Finding the y-intercept**
At the y-axis, \( x = 0 \). By substituting \( x = 0 \) into the function \( y = |\sqrt{0} - 2| \), we find \( y = |-2| = 2 \). Hence, the y-intercept is at the point (0, 2).

**Finding the x-intercept(s)**
For x-intercepts, \( y = 0 \). Setting the function equal to zero gives us \( |\sqrt{x} - 2| = 0 \). Solving this equation, we find \( \sqrt{x} = 2 \), and by squaring both sides, we conclude \( x = 4 \). Thus, the x-intercept is at the point (4, 0).

Remember, these intercepts are critical checkpoints in understanding how the function behaves and where it interacts with the axis.

Graphing a function requires plotting key points and drawing the curve that represents the function on a coordinate plane. The function \( y = |\sqrt{x} - 2| \) can be understood by grasping how each component affects the graph.

**Initial Sketch**
First, consider the graph of \( y = \sqrt{x} - 2 \). This graph typically starts at (0, -2) and slopes upward, passing through the x-intercept at (4, 0). However, the crucial aspect of graphing \( y = |\sqrt{x} - 2| \) is the absolute value function.

**Absolute Value Effect**
Anywhere \( \sqrt{x} - 2 \) would be negative, it is flipped over the x-axis by the absolute value operation. This creates a 'V' shape in the graph, as it reflects the curve above the x-axis.

**Key Points for Graphing**

• Start at the y-intercept (0, 2).
• Plot the x-intercept at (4, 0).
• Trace the mirrored graph to reflect over the y-axis symmetry.

This approach helps to accurately capture how the function interacts with the x-axis and ensures a correct graph.