Problem 20 - Algebra 2

Question

Graph each function. Identify the domain and range. \(f(x)=|2 x|\)

Step-by-Step Solution

Answer

The domain is \((-\infty, \infty)\) and the range is \([0, \infty)\).

Step 1 - Understanding the Function

The function given is the absolute value function \( f(x) = |2x| \). The absolute value function turns all values within the function to be non-negative.

Step 2 - Finding the Domain

The domain of a function is the set of all possible input values (x-values) the function can accept. Since there are no restrictions on \( x \) for the function \( f(x) = |2x| \), the domain is all real numbers, expressed as \( (-\infty, \infty) \).

Step 3 - Finding the Range

The range of a function is the set of all possible output values (y-values). For \( f(x) = |2x| \), since absolute value outputs non-negative values, the range is \( [0, \infty) \).

Step 4 - Graphing the Function

To graph \( f(x) = |2x| \), recognize it behaves like a V-shaped graph centered at the origin. For \( x > 0 \), the graph is a line \( y = 2x \), and for \( x < 0 \), the graph is \( y = -2x \). These lines meet at the origin (0,0).

Step 5 - Analyzing the Graph

The graph is symmetric about the y-axis, starting at the origin (0,0) and extending outward in both directions. Each side of the V has a steep slope, illustrating the effect of the factor of 2 in \( |2x| \).

Key Concepts

Domain and Range

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 Functions

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.

Piecewise Functions

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.