Problem 52 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understand Function Properties

A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain. It is odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition is satisfied for the same set of values, the function is neither even nor odd.

Step 2: Compare Values for Even Check

Compare \( f(x) \) and \( f(-x) \) for all values of \( x \) that are provided in the table:- \( f(-3) = -1 \) which is not equal to \( f(3) = 5 \).- \( f(-2) = 0 \) which is not equal to \( f(2) = 4 \).- \( f(-1) = 1 \) which is not equal to \( f(1) = 3 \).The function does not satisfy \( f(-x) = f(x) \) for any pair, suggesting it is not even.

Step 3: Compare Values for Odd Check

Now compare \( f(x) \) and \( -f(-x) \) for each \( x \):- \( f(-3) = -1 \) and \( -f(3) = -5 \), not equal.- \( f(-2) = 0 \) and \( -f(2) = -4 \), not equal.- \( f(-1) = 1 \) and \( -f(1) = -3 \), not equal.None of the values satisfy \( f(-x) = -f(x) \), meaning it is not odd.

Step 4: Conclude Function Type

Since the function is neither satisfying the conditions for being even nor odd, we conclude that the function is neither even nor odd.

Key Concepts

Function Properties

When discussing functions, we often explore their properties to understand their behavior. A couple of key properties are whether they are even or odd. Knowing these can help in graphing and predicting function behaviors.

• Even Functions: A function is called even if it satisfies the condition: \( f(-x) = f(x) \) for every value of \( x \). An easy way to spot an even function is by checking if its graph is symmetric with respect to the y-axis.
• Odd Functions: Conversely, a function is odd if it holds that \( f(-x) = -f(x) \) for all \( x \). If the graph of a function is symmetric about the origin, it indicates that the function is odd.

If neither of these properties holds, the function is considered neither even nor odd. Recognizing these traits can simplify mathematical processes, especially in calculus.

Function Symmetry

Symmetry in functions is a fascinating aspect that deals with how a function's graph behaves visually. Understanding this symmetry enhances our ability to interpret function properties more effortlessly.

• Y-axis Symmetry: This kind of symmetry indicates an even function. The y-axis acts like a mirror reflecting the two halves of the graph, such that each half is a mirror image of the other. This symmetry means \( f(x) = f(-x) \) at any point \( x \).
• Origin Symmetry: For odd functions, symmetry occurs around the origin. So if you rotate the graph 180 degrees around the origin, it maps onto itself. Mathematically, this means \( f(x) = -f(-x) \).

Understanding symmetry not only helps in identifying the type of function but also in graphing and transforming functions effortlessly.

Conjecture in Mathematics

A conjecture in mathematics is a proposition or hypothesis that is proposed based on preliminary observations, which seems true but has not yet been proven. To make a conjecture about the nature of a function (like whether it is even, odd, or neither), we typically analyze its properties and symmetry from a sample or limited number of data points - as seen from the given table in the exercise.

• By observing the patterns or lack thereof, we attempt to generalize about the entire function's behavior.
• If a conjecture cannot be definitively proven with the given information, further investigation or more comprehensive datasets might be needed.

In this exercise, after examining the table's numerical patterns, the conjecture was made that the function is neither even nor odd - a conclusion drawn after identifying that the symmetry conditions for both even and odd functions were not met. Constructing conjectures like this one helps drive deeper mathematical inquiry and exploration.