To determine whether the function [tex]\( f(x) = \frac{4x}{x^2 - 1} \)[/tex] is even, odd, or neither, we need to evaluate [tex]\( f(-x) \)[/tex] and compare it to [tex]\( f(x) \)[/tex] and [tex]\(-f(x)\)[/tex].
1. Calculate [tex]\( f(-x) \)[/tex]:
We start by substituting [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = \frac{4(-x)}{(-x)^2 - 1}
\][/tex]
2. Simplify [tex]\( f(-x) \)[/tex]:
Simplify the expression for [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = \frac{4(-x)}{x^2 - 1} = \frac{-4x}{x^2 - 1}
\][/tex]
3. Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
Now, let's compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
- [tex]\( f(x) = \frac{4x}{x^2 - 1} \)[/tex]
- [tex]\( f(-x) = \frac{-4x}{x^2 - 1} \)[/tex]
We observe that [tex]\( f(-x) \)[/tex] is the negative of [tex]\( f(x) \)[/tex]:
[tex]\[
f(-x) = -f(x)
\][/tex]
4. Conclusion:
Since [tex]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) \)[/tex] satisfies the condition for being an odd function.
Therefore, we conclude that the function [tex]\( f(x) = \frac{4x}{x^2 - 1} \)[/tex] is an [tex]\(\textbf{odd}\)[/tex] function.
