# Let  f(x)  be the function  f(x) = 5^x - 5^{-x}.  Is  f(x)  even, odd, or neither ? Prove your result.

Feb 6, 2018

The function is odd.

#### Explanation:

If a function is even, it satisfies the condition:

$f \left(- x\right) = f \left(x\right)$

If a function is odd, it satisfies the condition:

$f \left(- x\right) = - f \left(x\right)$

In our case, we see that

$f \left(- x\right) = {5}^{-} x - {5}^{x} = - \left({5}^{x} - {5}^{-} x\right) = - f \left(x\right)$

Since $f \left(- x\right) = - f \left(x\right)$, the function is odd.