Question

Determine algebraically whether \(f(x)=\frac{-x}{x^{2}+9}\) is even, odd, or neither.

Short answer

The function \(f(x)=\frac{-x}{x^2+9}\) is neither even nor odd.

Step-by-step solution

- Understand the Definitions

Recall that a function is even if \(f(-x) = f(x)\) for all x in its domain, and it is odd if \(f(-x) = -f(x)\) for all x in its domain. If neither condition is met, the function is neither even nor odd.

- Compute f(-x)

Substitute \(-x\) into the function \(f(x)\). \ f(-x) = \frac{-(-x)}{(-x)^2 + 9} = \frac{x}{x^2 + 9} \.

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

Compare \(f(-x) = \frac{x}{x^2 + 9}\) with \(f(x) = \frac{-x}{x^2 + 9}\). Clearly, \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\).

- Conclusion

Since \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\), the function \(f(x) = \frac{-x}{x^2 + 9}\) is neither even nor odd.

Key concepts

even functions

An even function is one where the function exhibits symmetry across the y-axis. This means that the function does not change when you replace x with -x.
Mathematically, this is written as:\[ f(-x) = f(x) \]
To determine if a function is even, replace every x with -x and simplify. If the function remains unchanged, it's even.
For example, the function \( f(x) = x^2 \) is even because:\[ f(-x) = (-x)^2 = x^2 = f(x) \]But in the given exercise, we found:\[ f(-x) = \frac{x}{x^2 + 9} eq f(x) = \frac{-x}{x^2 + 9} \]Thus, the function \( f(x) = \frac{-x}{x^2 + 9} \) is not even.

odd functions

An odd function exhibits symmetry about the origin, meaning that if you rotate it 180 degrees, it looks the same.
This can be described mathematically as:\[ f(-x) = -f(x) \]
To check if a function is odd, replace x with -x and see if the entire function equals the negative of the original function.
For example, the function \( f(x) = x^3 \) is odd because:\[ f(-x) = (-x)^3 = -x^3 = -f(x) \]In the given exercise, however, we have:\[ f(-x) = \frac{x}{x^2 + 9} eq -f(x) = -\frac{-x}{x^2 + 9} = \frac{x}{x^2 + 9} \]Therefore, our function \( f(x) = \frac{-x}{x^2 + 9} \) isn't odd.

function symmetry

Function symmetry is a concept in algebra that helps determine the behavior of functions under certain transformations. Even and odd functions are two specific types of symmetries.
For even functions, symmetry occurs around the y-axis.
For odd functions, symmetry occurs around the origin.
A function can also be neither even nor odd if it doesn't match either symmetry condition.
In the context of our given function \( f(x) = \frac{-x}{x^2 + 9} \), we tested both symmetries. We observed that it does not satisfy the conditions of either symmetry, leading us to conclude that the function is neither even nor odd.

algebra

Algebra involves the study of mathematical symbols and rules for manipulating those symbols. It is a unifying thread of almost all mathematics.
In this exercise, we used algebraic methods to identify the nature of function symmetry. By substituting -x into the given function, the algebraic manipulation revealed whether the function matched even or odd symmetry properties.
This simple but effective use of substitution allows us to explore the deeper properties of functions.

function properties

Function properties involve various characteristics that define how functions behave. Examples include:

• Domain and range
• Zeros of the function
• Symmetry (even or odd)
• Continuity
• Derivative and integral properties

In this exercise, we focused specifically on the symmetry property. By determining the symmetry, we gain insights into the function's behavior and graphical representation.
For instance, knowing a function is even or odd helps simplify the analysis of its integral over symmetric limits.