Is this function injective,surjective?

Question

I have a function: $F:R\rightarrow R^+,f(x)=a ^{\lvert x\rvert},0<a<1$. Is this function surjective,injective?I know,that function is injective if $f(x_1)=f(x_2) \Rightarrow x_1=x_2$.My function won't be injective,since both $a$ and $x$ are always positive.Is my judgement correct?I know a function is surjective if each element of the codomain $y\in R$ is the image of some element in the domain $x\in R^+$,I tried finding the inverse of the function but failed.What can I do here?

Answer 1

$ f $ is not injective since

$$f(-1)=f(1)=a$$

$ f $ is not sutjective because

$$(\forall x\in \Bbb R) \; f(x)=e^{|x|\ln(a)}$$

with $$ \ln(a)<0$$ so, $$(\forall x\in \Bbb R)\; f(x)\le 1$$ and if we take for example $ y=2 $, there will be no $ x\in \Bbb R $ such that $f(x)=y$.