The domain of $f(x) = a^x + a^{-x}, a>0$.
My attempt :
Since the domain of the exponential function is $\mathbb{R}$, and since our function is the sum of two exponential functions and since the exponential function never attains $0$ value, then the domain of our function is $\mathbb{R}$. am I correct?
And it is an even function.correct?
Yes, the domain is $\mathbb R$. But the fact that the exponential function never attains $0$ is not relevant here. All you need to know is that, for any real number $x$, both $a^x$ and $a^{-x}$ are defined.
And, yes, it is even: $f(-x)=a^{-x}+a^x=a^x+a^{-x}=f(x)$.