Understanding the Properties of the Function f(x) = 2|x| | A Breakdown and Analysis

f(x) = 2|x|

The given function is f(x) = 2|x|

The given function is f(x) = 2|x|. Let’s break down the function and understand its properties.

The function f(x) = 2|x| involves an absolute value symbol, which denotes the magnitude or distance of a number from zero without considering its sign. The absolute value of a number x is denoted as |x|.

In this case, the absolute value of x is multiplied by 2, represented as 2|x|. Multiplying any value by 2 simply doubles the value.

To understand how this function behaves, we can analyze two cases: when x is positive and when x is negative.

1) When x is positive:
For positive values of x, the absolute value |x| is equal to x itself. Therefore, for x > 0, f(x) = 2x. This means that the function behaves linearly with a positive slope for positive x values. As x increases, the output f(x) increases linearly twice as fast.

2) When x is negative:
For negative values of x, the absolute value |x| is equal to -x. Therefore, for x < 0, f(x) = 2(-x), which simplifies to -2x. This means that the function behaves linearly with a negative slope for negative x values. As x becomes more negative, the output f(x) increases, but in the opposite direction. In summary, the function f(x) = 2|x| creates a V-shaped graph with the vertex at the origin (0, 0). It is a piecewise linear function with a positive slope for positive x values and a negative slope for negative x values.

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