Question

How do you determine if `f(x) = 2^x + 2^-x` is an even or odd function?

Answer 1

`f(x)=2^x+2^(-x)` is an even function;
but not an odd function.

Explanation:

even function definition
`f(x)` is an even function if `color(black)(f(x)=f(-x))` for all `x` in the domain of `f`.

`f(x)=2^color(red)(x)+2^color(blue)(-x)`
`f(-x)=2^color(red)(-x)+2^color(blue)(-(-x))=color(blue)(2^(x)+2^color(red)(-x)`
`f(x)=f(-x) rArr f(x)` is even.

odd function definition
`f(x)` is and odd function if `color(black)(f(-x)=-f(x)` for all `x` in the domain of `f`.

if `x=1`
`f(-x=-1)=2^(-1)+2^-(-1)=1/2+2 =2 1/2`
`-f(x=1) = -(2^1+2^-1) = -2 1/2`
`f(-x)!=-f(x)` (when `x=1`) `rArr f(x)` is not an odd function