Understanding the Absolute Value Function: Exploring the V-Shaped Graph and Piecewise Nature

f(x) = |x|

The function f(x) = |x| represents the absolute value of x

The function f(x) = |x| represents the absolute value of x. The absolute value of a number is its distance from zero on a number line, regardless of its sign.

To understand the function f(x) = |x|, let’s consider the two possible cases separately:

1. When x is positive or zero (x ≥ 0):
In this case, the absolute value of x is equal to x itself since x is already a positive or zero value. So, f(x) = x.

2. When x is negative (x < 0): When x is negative, the absolute value of x is obtained by taking the negative value of x. So, f(x) = -x. Graphically, the function f(x) = |x| creates a V-shaped graph, also known as a V-curve. The point (0,0) lies at the vertex of the V-curve. On the left-hand side of the vertex (the negative x-axis), the graph follows the line with a slope of -1. On the right-hand side of the vertex (the positive x-axis), the graph follows the line with a slope of 1. For example, if we evaluate f(x) for various values of x: - f(2) = |2| = 2 - f(-3) = |-3| = 3 In summary, the function f(x) = |x| is a piecewise function that gives the absolute value of a number x. It returns x for non-negative values of x and -x for negative values of x. The graph of this function is a V-curve.

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