Part 3/5:
The absolute value function can be classified as a piecewise function due to its two distinct behaviors based on the sign of ( x ). For more complex formulations involving absolute values, piecewise functions become essential.
Example: Absolute Value of a Quadratic Function
Let's consider ( y = |x^2| ). Notably, since ( x^2 ) is always non-negative (0 or positive), the absolute value simply returns ( x^2 ):
[
y = |x^2| = x^2
]
The graph of this function will be a parabola that opens upwards, consistent in being positive for all values of ( x ), reflecting the characteristic of quadratic functions.
Example: Absolute Value of a Cubic Function
When dealing with the cubic function ( y = |x^3| ), the scenario slightly shifts due to the nature of cubic numbers.