91Ó°ÊÓ

When we talk about the domain of a function, we mean the set of all potential inputs, or x-values. For the absolute value function \( f(x) = |2x| \), the domain is unrestricted. You can plug any real number into \( 2x \), apply the absolute value, and get a valid result. So, the domain is all real numbers, written as \( (-\infty, \infty) \).

Next, let's discuss the range, which is all possible outputs, or y-values, that the function can produce. Since the absolute value always turns results non-negative, \( f(x) = |2x| \) will only output values equal to or greater than zero. Consequently, the range of this function is \( [0, \infty) \). This means y can be zero or any positive number.

Graphing an absolute value function like \( f(x) = |2x| \) can be a straightforward process. The absolute value function creates a graph with a distinctive V-shape.

First, consider the behavior of the function on either side of the y-axis. For values of \( x \) greater than zero, the function behaves like \( y = 2x \), a line with a positive slope of 2. This means it rises quickly as you move to the right.

For \( x \) less than zero, the function resembles \( y = -2x \). This line descends quickly as it moves to the left.

Both lines meet at the origin (0,0), providing symmetry around the y-axis. This symmetry is crucial for understanding how the graph is centered and how steeply it rises. The factor of 2 affects the V's openness by steepening the arms of the V-shape.

The concept of piecewise functions is beneficial when dealing with absolute value expressions like \( f(x) = |2x| \). A piecewise function breaks down a single function into different 'pieces' based on intervals of the input value.

For \( f(x) = |2x| \), it can be expressed as a piecewise function:

• When \( x \geq 0 \), the function is \( f(x) = 2x \).
• When \( x < 0 \), the function is \( f(x) = -2x \).

This breakdown helps us understand the absolute value by clearly dividing the graph into its positive and negative sections.

Using piecewise notation gives us a clearer picture of how a function like this behaves across different domains of x, showing exactly where one part of the function ends and another begins.