To determine whether a given function is even, odd, or neither, we will use the following definitions:
- Even Function: [tex]\( f(x) = f(-x) \)[/tex]
- Odd Function: [tex]\( f(x) = -f(-x) \)[/tex]
Given the function [tex]\( f(x) = \frac{2x^2}{x+1} \)[/tex]:
1. Evaluate [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = \frac{2x^2}{x + 1}
\][/tex]
2. Substitute [tex]\(-x\)[/tex] into the function to find [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = \frac{2(-x)^2}{-x + 1} = \frac{2x^2}{-x + 1}
\][/tex]
3. Simplify and compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex].
Now, we have [tex]\( f(x) = \frac{2x^2}{x + 1} \)[/tex] and [tex]\( f(-x) = \frac{2x^2}{-x + 1} \)[/tex].
4. Let's see if [tex]\( f(x) = f(-x) \)[/tex]:
[tex]\[
f(x) = \frac{2x^2}{x + 1} \not= f(-x) = \frac{2x^2}{-x + 1}
\][/tex]
Since [tex]\( f(x) \)[/tex] is not equal to [tex]\( f(-x) \)[/tex], the function is not even.
5. Next, check if [tex]\( f(x) = -f(-x) \)[/tex]:
[tex]\[
-f(-x) = -\left( \frac{2x^2}{-x + 1} \right) = \frac{-2x^2}{-x + 1} = \frac{-2x^2}{x - 1}
\][/tex]
Since [tex]\( f(x) = \frac{2x^2}{x + 1} \not= \frac{-2x^2}{x - 1} \)[/tex], the function is not odd.
6. Conclusion:
The function [tex]\( f(x) = \frac{2x^2}{x + 1} \)[/tex] is neither even nor odd, since it does not satisfy the conditions for being either even or odd.
Hence, after evaluating and comparing [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex], we conclude that the function is neither even nor odd.
