Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam Determine whether the function \(f(x)=\frac{\sqrt[3]{x}}{x^{2}+6}\) is even, odd, or neither.

Short answer

The function is odd.

Step-by-step solution

- Understand the Definitions

A function is even if for all x in the domain of f, we have: f(-x) = f(x). A function is odd if for all x in the domain of f, we have: f(-x) = -f(x). If neither of these conditions hold, the function is neither even nor odd.

- Find f(-x)

To determine whether the function is even or odd, we need to find f(-x) and compare it to f(x) and -f(x). Given f(x) = \( \frac{\sqrt[3]{x}}{x^2 + 6} \), let's find f(-x): f(-x) = \( \frac{\sqrt[3]{-x}}{(-x)^2 + 6} \).

- Simplify f(-x)

First, simplify the numerator: \( \sqrt[3]{-x} = -\sqrt[3]{x} \). Next, simplify the denominator: \( (-x)^2 = x^2 \). So, f(-x) = \( \frac{-\sqrt[3]{x}}{x^2 + 6} \).

- Compare f(-x) to f(x) and -f(x)

We now compare f(-x) = \( \frac{-\sqrt[3]{x}}{x^2 + 6} \) to f(x) = \( \frac{\sqrt[3]{x}}{x^2 + 6} \) and -f(x) = \( -\frac{\sqrt[3]{x}}{x^2 + 6} \). Clearly, f(-x) = -f(x).

- Conclusion

Since f(-x) = -f(x), the function f(x) = \( \frac{\sqrt[3]{x}}{x^2 + 6} \) is odd.

Key concepts

Even Functions

Even functions have a special type of symmetry. For a function to be even, its graph must be symmetric with respect to the y-axis. This means, for every point \(x, f(x)\), there is a corresponding point \(-x, f(x)\). Mathematically, a function \(f(x)\) is even if \(f(-x) = f(x)\).

Here are some key properties of even functions:

• If you fold the graph of the function along the y-axis, the two halves will match perfectly.
• Even functions include examples like \(f(x) = x^2\) and \(f(x) = \cos(x)\).
• If a function is even, its output remains the same whether you input \(x\) or \(-x\).

By applying this concept to the given function \(f(x)=\sqrt[3]{x}/(x^2+6)\), if we replace \(x\) with \(-x\)\, we should get an expression identical to the original function for it to be even. However, we observe that \(f(-x) = -f(x)\), i.e., the function is not even.

Function Symmetry

Function symmetry is all about understanding how a function's graph behaves relative to certain axes or points. Two primary symmetries are even and odd symmetries:

• Even Symmetry: As mentioned, even functions are symmetric along the y-axis. For any \(x\) in the domain, \(f(-x) = f(x)\).
• Odd Symmetry: For a function to have odd symmetry, the graph must be symmetric with respect to the origin. In mathematical terms, a function \(f(x)\) is odd if \(f(-x) = -f(x)\). If you rotate the graph 180 degrees around the origin, it will match up with itself.

For the function \(f(x)=\sqrt[3]{x}/(x^2+6)\), when replacing \(x\) with \(-x\), we compared \(-f(x)\). Since \(f(-x) = -f(x)\), it confirms that the function has odd symmetry. Therefore, function symmetry helps identify the type of function by examining such transformations.

Function Transformations

Understanding function transformations is crucial to grasping how graphs change. Transformations include shifts, stretches, compressions, and reflections.

• Shifts: Horizontal and vertical shifts move the graph left, right, up, or down.
• Stretches and Compressions: Functions can be stretched or compressed vertically or horizontally by multiplying by a constant.
• Reflections: A function can be reflected over the x-axis or y-axis. For example, \(f(-x)\) represents a reflection over the y-axis, and \(-f(x)\) represents a reflection over the x-axis.

When we examined \(f(x)=\sqrt[3]{x}/(x^2+6)\) by replacing \(x\) with \(-x\)\ and found \(-f(x)\), it involved reflection about the origin. This helped us conclude the function is odd. Understanding these transformations allows students to identify changes in the graph and recognize the function's behavior under different inputs.