Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\-3 & -1 \\\\-2 & 0 \\\\-1 & 1 \\\0 & 2 \\\1 & 3 \\\2 & 4 \\\3 & 5\end{array}$$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understand the Definition of Even and Odd Functions

An even function satisfies the condition \( f(x) = f(-x) \) for all \( x \) in the domain of \( f \). An odd function satisfies \( f(x) = -f(-x) \) for all \( x \). A function that does not meet either of these criteria is neither even nor odd.

Step 2: Check for Even Function

To determine if the function is even, compare \( f(x) \) with \( f(-x) \) for each pair of \( x \) values.- For \( x = -3 \) and \( x = 3 \), \( f(x) = -1 \) and \( f(-x) = 5 \), so \( f(x) eq f(-x) \).- For \( x = -2 \) and \( x = 2 \), \( f(x) = 0 \) and \( f(-x) = 4 \), so \( f(x) eq f(-x) \).- For \( x = -1 \) and \( x = 1 \), \( f(x) = 1 \) and \( f(-x) = 3 \), so \( f(x) eq f(-x) \).This indicates the function is not even.

Step 3: Check for Odd Function

To determine if the function is odd, check if \( f(x) = -f(-x) \) for each \( x \).- For \( x = -3 \) and \( x = 3 \), \( f(x) = -1 \) and \( -f(-x) = -5 \), so \( f(x) eq -f(-x) \).- For \( x = -2 \) and \( x = 2 \), \( f(x) = 0 \) and \( -f(-x) = -4 \), so \( f(x) eq -f(-x) \).- For \( x = -1 \) and \( x = 1 \), \( f(x) = 1 \) and \( -f(-x) = -3 \), so \( f(x) eq -f(-x) \).This indicates the function is not odd.

Step 4: Conclusion

Since the function is neither satisfying the property for even functions nor for odd functions, we conclude that the function \( f \) is neither even nor odd.

Key Concepts

Even Functions

Even functions have a very symmetrical property when it comes to their graphs. This symmetry is around the vertical y-axis. A mathematical way to express this symmetry is through the equation: \( f(x) = f(-x) \). This means that if you pick any point \( x \) on the function, the value of \( f(x) \) should be the same as \( f(-x) \), which is the point directly opposite across the y-axis.Here are some characteristics of even functions:

• Their graphs reflect perfectly over the y-axis.
• If one point \( (x, f(x)) \) is on the graph, the point \( (-x, f(x)) \) will also be on the graph.
• Common examples of even functions are \( x^2 \), \( \cos(x) \), and \( |x| \).

When checking a table of values, if \( f(x) eq f(-x) \) for any \( x \), the function is not even. This lack of symmetry was observed in the exercise's table, confirming the function is not even.

Odd Functions

Odd functions also feature symmetry, but it's a bit different. Their symmetry is rotational, specifically around the origin. The defining mathematical characteristic is \( f(x) = -f(-x) \).What this means is: if you take a point \( x \) from the function and look at \( f(x) \), then \(-x\) will have \(-f(x)\). Visually, this is like flipping the function 180 degrees around the origin.Key features of odd functions include:

• The graphs have rotational symmetry about the origin, meaning they look the same if rotated halfway around the origin.
• If the point \( (x, f(x)) \) is on the graph, the point \( (-x, -f(x)) \) should also be on the graph.
• Examples of odd functions include \( x^3 \), \( \sin(x) \), and the function \( x \) itself.

In the exercise's table, we don't find \( f(x) = -f(-x) \) consistently. This confirms that the function is not odd.

Function Analysis

Function analysis helps us understand the nature and behavior of functions. When analyzing a function, determining if it's even or odd is a crucial part of understanding its symmetry and how it may look graphically.The analysis involves:

• Calculating and comparing \( f(x) \) with \( f(-x) \) for evenness.
• Checking \( f(x) = -f(-x) \) for oddness.
• If neither condition is met, the function could possess neither symmetry and might be called 'neither even nor odd'.

In our simple dataset exercise, analyzing values clarified that the function exhibits neither type of symmetry. It's important to remember this method can be applied universally to check for symmetry in any function.