Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=|x+2|$$

Step-by-Step Solution

Answer

The function \(f(x) = |x+2|\) is neither even nor odd.

Step 1: Graph the function

The first step is to graph the function. For a function \(f(x) = |x+2|\), it will result to a V-shaped graph. The vertex of this graph will be at (-2,0), since the absolute value function \(|x+2|\) shifts the vertex of the basic absolute value function \(|x|\) 2 units to the left. To visualize this, use a graphing utility or calculator and plot the graph of \(f(x) = |x+2|\).

Step 2: Examine the graph

Look at the graph and determine if it is symmetric about the y-axis or the origin. The graph is symmetric about the line x = -2 (which is a vertical line), not about the y-axis or the origin. Therefore, it is not even, nor is it odd.

Step 3: Mathematical verification

To verify, substitute \(x\) with \(-x\) in the original equation and see if it equals to \(f(x)\) or \(-f(x)\). The function \(f(-x) = |-x + 2|\) does not equal to either \(f(x)\) or \(-f(x)\). So the function is neither even nor odd.

Key Concepts

Absolute Value

The concept of absolute value is quite crucial in understanding various mathematical functions. Essentially, the absolute value of a number is its distance from zero on the number line, regardless of the direction. For example, the absolute value of both 3 and -3 is 3.

Absolute value functions often produce a distinct V-shape when graphed. This is due to how absolute value affects both positive and negative inputs, always turning them into non-negative outputs. Consider the function \(f(x) = |x+2|\). In this case, the graph has a vertex at (-2,0) and opens upwards.

To graph absolute value functions, follow these steps:

• Identify the vertex of the function. For \(|x+2|\), the graph shifts the basic \(|x|\) 2 units to the left.
• Check the slope which is often 1 or -1 for simple absolute value functions.
• Plot points on either side of the vertex to outline the V-shape, ensuring symmetry is maintained.

Using these principles, you can decipher the graph and behavior of any basic absolute value function with ease.

Symmetry

Symmetry in functions is a fascinating aspect that helps in determining if the function is even, odd, or neither. Generally, symmetry can be about the y-axis, the origin, or a specific line. However, not all functions exhibit symmetry in these conventional ways.

For functions, there are two primary types of symmetry:

• **Y-axis Symmetry:** This indicates that if you fold the graph along the y-axis, both halves would match. This typically means the function is even.
• **Origin Symmetry:** If rotating the graph 180 degrees about the origin results in the same graph, the function is considered odd.

In the given function \(f(x) = |x+2|\), instead of reflecting over the y-axis or the origin, it is symmetric about the vertical line \(x = -2\). This non-standard symmetry does not qualify it as an even or odd function.

One should be mindful that most textbook examples of symmetry focus on traditional axes. Exploring non-standard lines of symmetry, like in this case, is beneficial for a complete mathematical understanding.

Even and Odd Functions

Even and odd functions are classified based on their symmetry properties and how they behave when inputs are negated. This property becomes particularly pivotal when analyzing and graphing functions.

**Even Functions** are symmetric about the y-axis. Mathematically, a function \(f(x)\) is even if \(f(-x) = f(x)\) for all x in the function's domain. This makes the function's graph identical on both sides of the y-axis.

**Odd Functions** have origin symmetry. This means \(f(-x) = -f(x)\) for all x. When graphed, the function and its rotated 180-degree version around the origin look the same.

For the function \(f(x) = |x+2|\):

• Substituting \(-x\) gives \(|-x+2|\), which is not equal to either \(f(x)\) or \(-f(x)\).
• This confirms that \(f(x)\) is neither even nor odd.

Being able to discern and test for these properties manually ensures a robust understanding of the nature of functions beyond just graphing utility outputs.