show if function is even or odd

Suppose that we have equation: $$f(x)=\frac{2^x+1}{2^x-1}$$

There is question if this function even or odd? I know definitions of even and odd functions, namely even is if $f(-x)=f(x)$ and odd is if $f(-x)=-f(x)$ and when I put $-$ sign in function, found that this is neither even nor odd function, because $2^{-x}\ne-1 \times 2^x$, but my book says that it is even, so am I wrong? Please help me to clarify book is correct or me? Thanks

• As this example shows, in order to show that a function is neither even or odd, it is not enough simply to invert the variable and get something that looks different. The only reliable way to conclude neither-even-nor-odd is to find a concrete number such that you can plug it and its negative into the function and get results that are neither equal or opposite. – Henning Makholm Jun 17 '12 at 14:18
• This example also shows that, sometimes, books can be wrong! This is not an even function, as almost every answer below states explicitly. – Cameron Buie Jun 17 '12 at 15:43
• You can always graph the function and see that it's definitely not even, that it in fact looks like it's odd (something others have proved below). – Ken Bloom Jun 17 '12 at 17:02

Let's see what $f(-x)$ looks like:

$$f(-x) = \frac{2^{-x} + 1}{2^{-x} - 1}$$

Since $f(x)$ contains $2^x$ and not $2^{-x}$, let's multiply the numerator and denominator by $2^x$:

$$f(-x) = \frac{2^x(2^{-x} + 1)}{2^x(2^{-x} - 1)} = \frac{1 + 2^x}{1-2^x} = - \frac{2^{x} + 1}{2^{x} - 1} = -f(x)$$

This shows that the function is odd.

The function is odd. You propably miss something in your calculation.

$\displaystyle{ f(-x) = \frac{2^{-x}+1}{2^{-x}-1}= \frac{ \frac{1}{2^x} + 1 }{ \frac{1}{2^x} -1} = \frac{ \frac{1+2^x}{2^x}}{ \frac{1-2^x}{2^x}} =-f(x)}$

• yes i was thinking,that neither even or odd is this function,so have not finished calculation – dato datuashvili Jun 17 '12 at 13:47

$$\frac{2^{-x}+1}{2^{-x}-1} = \frac{\frac 1 {2^x}+1}{\frac 1 {2^x}-1}$$

Now clear factions by multiplying numerator and denominator by $2^x$ and see what you have.

We get that $$f(-x) = \frac{2^{-x}+1}{2^{-x}-1} = \frac{1+2^x}{1-2^x}= - \frac{2^x + 1}{2^x-1} = -f(x),$$ so the function is odd.