Problem 10 - Precalculus : Building Concepts and Connections

Question

Decide if each function is odd, ezen, or neither by using the appropriate definitions. $$\begin{array}{cccccc}x & -3 & -1 & 0 & 1 & 3 \\\f(x) & -5 & -7 & -10 & -7 & -5\end{array}$$

Step-by-Step Solution

Answer

The provided function is an even function.

Step 1: Check for Even Function

For a function to be even, the condition \(f(x) = f(-x)\) has to be satisfied. From the provided values, we can test this by checking if the f(1) equals f(-1), and also if f(3) equals to f(-3). As we substitute in the corresponding f(x) values from the table, f(1) = -7 while f(-1) = -7 and f(3) = -5 while f(-3) = -5. Therefore, the function satisfies the condition for being even.

Step 2: Check for Odd Function

For a function to be odd, the condition \(f(-x) = -f(x)\) must be satisfied. Let's test this with the values given in the table. By substitution, f(-1) = -f(1) would imply -7 = -(-7) and f(-3) = -f(3) would imply -5 = -(-5). However, the second part of each of these statements is not true since these equal 7 and 5 respectively instead of -7 and -5. Therefore, the condition for the function to be odd is not met.

Key Concepts

Even Functions

Understanding even functions is essential for gaining insights into the symmetry aspects of mathematics. An even function exhibits a specific type of symmetry with respect to the y-axis.

An even function is defined by the condition that for every value of x in the function's domain, the equation \(f(x) = f(-x)\) holds true. Visually, this means if you were to fold the graph along the y-axis, both halves would match up perfectly, much like the two sides of a butterfly’s wings.

Here's a simple way to check whether a function is even:

• Pick a positive value of x.
• Calculate \(f(x)\) and \(f(-x)\).
• If these values are equal, that condition holds for all correspondingly positive and negative values of x; your function is even.

From our textbook exercise, we found that the function satisfied the even function condition, as corresponding positive and negative x-values yielded the same f(x). To solidify understanding, let's consider the classic even function, \(f(x) = x^2\). No matter what x-value you choose, squaring it will give you the same result as squaring its negative counterpart, reinforcing the definition of an even function.

Odd Functions

In contrast to even functions, odd functions have a different kind of symmetry – rotational symmetry about the origin. For a function to be classified as odd, it must satisfy the condition \(f(-x) = -f(x)\) for every x in its domain.

Intuitively, if you rotate the graph of an odd function 180 degrees about the origin, you will get the same graph back. This occurs because the output for a positive input is the mirror image through the origin of the output for the corresponding negative input.

To determine if a function is odd, you can use these steps:

• Select a positive x-value.
• Compute both \(f(x)\) and \(f(-x)\).
• The function is odd if \(f(-x)\) is the negative of \(f(x)\), for all pairs of x and -x.

In our textbook example, we checked for oddness but found the condition was not met, illustrating that the function in the exercise is not an odd function. An example of an odd function is \(f(x) = x^3\). Here, when you input a negative x-value, the cube of that number will be the negative of the cube of the positive x-value, satisfying the criteria for an odd function.

Function Symmetry

The concept of function symmetry revolves around the idea that certain functions mirror themselves across a line or a point. There are two primary types of symmetrical functions that we often study: even and odd functions, which showcase symmetry about the y-axis and the origin, respectively.

Function symmetry aids in understanding the behavior of graphs and in solving equations graphically. Additionally, symmetrical properties often have practical applications in physics and engineering where waveforms and signals can exhibit symmetry.

It’s worth noting that not all functions are symmetrical. When a function doesn't meet the criteria for being even or odd, we simply say that the function is 'neither even nor odd'. A function with no symmetry can still be analyzed and graphed, and it can also model many real-world phenomena where symmetry is not present.

In summary, recognizing even and odd functions helps in graphing, simplifies certain types of problems, and can reveal insights into the inherent nature of various mathematical relationships. The function from our exercise, verified as even, showed us an instance of y-axis symmetry. Always keep in mind that the quest for symmetry in functions is not just a pursuit of beauty in mathematics, but a practical tool for analysis and problem-solving.