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 0 on a number line, always resulting in a non-negative value.
When x is positive or zero (x ≥ 0), the absolute value of x is equal to x. So, f(x) = x for x ≥ 0.
When x is negative (x < 0), the absolute value of x is equal to the negation of x. So, f(x) = -x for x < 0. In other words, the function f(x) = |x| takes any real number x as input and returns its non-negative value. Let's visualize this function on a coordinate plane. Consider the x-axis as the input and the y-axis as the output: For x ≥ 0, the graph of f(x) = x is a diagonal line starting from the origin (0,0) that goes through all points in the first quadrant, rising at a 45-degree angle. It represents the positive values of x. For x < 0, the graph of f(x) = -x is the reflection of the above line in the y-axis. It starts from the origin (0,0) and goes through all points in the second quadrant, falling at a 45-degree angle. It represents the negative values of x. Therefore, the graph of f(x) = |x| is a "V"-shaped graph that opens upwards, with its vertex at the origin (0,0).
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