Problem 57 - 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. $$\begin{array}{r|r}x & f(x) \\\\-3 & 5 \\\\-2 & 4 \\\\-1 & 3 \\\0 & 2 \\\1 & 1 \\\2 & 0 \\\3 & -1\end{array}$$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understanding Odd and Even Functions

To determine whether a function is even or odd based on its values, recall that an **even function** satisfies \( f(-x) = f(x) \) for all \( x \), and an **odd function** satisfies \( f(-x) = -f(x) \) for all \( x \). A function that does not meet these criteria for all \( x \) is neither even nor odd.

Step 2: Evaluating f(x) for Evenness

Check each pair of values \( f(-x) \) and \( f(x) \) from the table to see if \( f(-x) = f(x) \).\[- f(-3) = 5 \quad eq f(3) = -1\- f(-2) = 4 \quad eq f(2) = 0\- f(-1) = 3 \quad eq f(1) = 1\]\(-0\) is always its opposite, hence \( f(-0) = f(0) = 2 \). Since at least one pair does not satisfy \( f(-x) = f(x) \), the function is not even.

Step 3: Evaluating f(x) for Oddness

Check each pair of values \( f(-x) \) and \( -f(x) \) from the table to see if \( f(-x) = -f(x) \).\[- f(-3) = 5, \quad -f(3) = -(-1) = 1 \- f(-2) = 4, \quad -f(2) = -(0) = 0 \- f(-1) = 3, \quad -f(1) = -(1) = -1\]None of these satisfy \( f(-x) = -f(x) \) for each pair, so the function is not odd.

Step 4: Conclusion

Since the function does not satisfy the condition for being even or odd across the domain, \( f(x) \) is neither even nor odd. No consistent relationship holds for either \( f(-x) = f(x) \) or \( f(-x) = -f(x) \) across all values in the table.

Key Concepts

Function Properties

The properties of a function are essential to understanding its behavior and characteristics. An even function has the property that the function's output is the same for input values at equal distances from the origin, just in opposite directions. Formally, for a function to be considered even, it must satisfy the condition: - \( f(-x) = f(x) \) for all values of \( x \) in its domain.
On the other hand, an odd function exhibits symmetry such that when the input value is negated, the output is also negated. This property is defined as:- \( f(-x) = -f(x) \) for all \( x \).
Functions that don’t meet either of these criteria in their entire domain are classified as neither even nor odd. Understanding these properties helps us in the study of function symmetry, which is crucial for analyzing graphs and predicting function behavior based on known inputs.

Mathematical Conjecture

In mathematics, making conjectures is a significant part of the problem-solving process. A conjecture is essentially an educated guess that is based on observations and patterns in data. When we examine the ordered pairs from the exercise table, we aim to determine if the function is even, odd, or neither by using the properties discussed earlier.
To form a conjecture, one should:

• Analyze symmetry in the function's values.
• Check for consistency across the set of ordered pairs.
• Apply logical reasoning to predict a pattern or rule.

In the case of the exercise, the table of values did not fit neatly into the definitions of either an even or odd function, leading to the conjecture that the given function is neither. Forming such conjectures helps in building deeper insights before moving to formal mathematical proofs.

Ordered Pairs Analysis

Analyzing ordered pairs is a practical method to understand the behaviors and properties of a function. Each pair consists of an input value \( x \) and an output value \( f(x) \). By examining how these pairs relate, we can detect patterns and symmetry which are indicators of specific function properties.
Here’s how to perform an ordered pair analysis:

• Compare each pair with its counterpart \( (-x, f(-x)) \).
• Verify against standard properties, such as evenness \( f(-x) = f(x) \) and oddness \( f(-x) = -f(x) \).
• Note any discrepancies or consistencies in the values.

This process involves not just looking at the numbers, but comprehending the symmetry or lack thereof in the function's output. The analysis from the exercise reveals that no consistent pattern of function symmetry (even or odd) exists, thus, the function is neither.