Problem 51 - 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. $$\begin{array}{r|r} x & f(x) \\ \hline-3 & 5 \\ -2 & 4 \\ -1 & 3 \\ 0 & 2 \\ 1 & 1 \\ 2 & 0 \\ 3 & -1 \end{array}$$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understand Even and Odd Functions

An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \) in its domain, meaning it is symmetric with respect to the y-axis. An odd function satisfies the condition \( f(-x) = -f(x) \) for all \( x \) in its domain, meaning it is symmetric with respect to the origin.

Step 2: Compare Values for Symmetry Tests

We need to compare each pair of \( f(x) \) and \( f(-x) \) from the table to check if \( f(x) = f(-x) \) or \( f(x) = -f(-x) \).

Step 3: Check Even Function Condition

For the ordered pair \((-3, 5)\), \( f(3) = -1 \). Since \( 5 eq -1 \), the function does not satisfy \( f(x) = f(-x) \). Repeat for \((-2, 4)\) and \((2, 0)\): \( 4 eq 0 \). For all other symmetric pairs, both conditions are not satisfied.

Step 4: Check Odd Function Condition

For the ordered pair \((-3, 5)\), \( f(3) = -1 \). Since \( 5 eq -(-1) \), the function does not satisfy \( f(x) = -f(-x) \). Repeat for \((-2, 4)\) and \((2, 0)\): \( 4 eq -0 \). The function does not satisfy the odd condition either.

Step 5: Conclude Based on Symmetry Checks

Since the function does not satisfy either the even or odd function conditions for any symmetric pair of \( x \) values in the table, it is concluded that the function is neither even nor odd.

Key Concepts

Even Functions

Even functions hold a special property: when you replace every x with -x, the output remains unchanged. Mathematically, it's expressed as \( f(-x) = f(x) \) for all x in the domain of the function. This property translates to symmetry about the y-axis. Imagine folding the graph along the y-axis — if both halves match perfectly, it’s an even function.

Here are some characteristics of even functions:

• Symmetrical about the y-axis.
• If \( (x, y) \) is a point on the graph, then \( (-x, y) \) is also a point.
• Common examples include \( f(x) = x^2 \) or \( f(x) = \cos(x) \).

In the exercise, we checked pairs like \((-3, 5)\) and \((3, -1)\). As \(5 eq -1\), the function failed the even test. Thus, it doesn't possess even function symmetry, confirming it's not an even function.

Odd Functions

Odd functions exhibit a distinct behavior: flipping both the input and the output of the function gives you the original function with a sign change. In simple terms, \( f(-x) = -f(x) \). These functions are symmetric around the origin of the graph.

Visualizing symmetry in odd functions:

• If you rotate the graph 180 degrees around the origin, it looks the same.
• A point \((x, y)\) on the curve indicates a point \((-x, -y)\) as well.
• Examples are \( f(x) = x^3 \) or \( f(x) = \sin(x) \).

In our step-by-step exploration, testing pairs like \((-3, 5)\) and its counterpart \((3, -1)\) showed that \(5 eq -(-1)\). Therefore, the function does not satisfy the odd function criterion, confirming it lacks this symmetry.

Symmetry

Symmetry in functions can provide insights into their properties and behaviors. Understanding symmetry types helps determine if a function is even, odd, or neither.

Essential types of symmetry:

• Y-axis symmetry: Typical of even functions, resulting in reflection across the y-axis.
• Origin symmetry: Characteristic of odd functions, involves a rotational symmetry of 180 degrees around the origin.
• No symmetry: When neither conditions for even nor odd functions are met, the function exhibits no symmetry.

The table from the exercise revealed that neither y-axis nor origin symmetry applies, leading to the conclusion that the function is neither even nor odd. Such insights can guide us to better understand functions, predict their graphs, and solve equations involving them.