Which of the following functions is a one-to-one and onto function from real numbers to real numbers (R to R).

$$\ln\left(e^{|x|}\right)$$
at x = a, the value of $$\ln\left(e^{|x|}\right)$$ is equal to  $$\ln\left(e^{a}\right)$$
at x = -a, the value of $$\ln\left(e^{|x|}\right)$$ is equal to $$\ln\left(e^{a}\right)$$
The function is not one-one for (R to R).

$$e^{\ln\left(x\right)}$$ is defined only on the set of positive numbers because negative numbers are not in the domain of logarithmic functions , so not one-one onto for (R to R).

Similarly we can eliminate $$e^{\ln\left(|x|\right)}$$ due to the same logic.

$$\ln\left(e^{x}\right)$$
It is defined on the entire set (R to R).
It will be one-one if $$f\left(x_1\right)=f\left(x_2\right)$$ implies $$x_1=x_2$$
$$\ln\left(e^{x}\right)$$ = $$x$$ for all $$x$$ in (R to R). So it is a one-one function.
It will be an onto function if every element in the codomain is covered.
$$e^x$$ is a monotonically increasing function throughout R and is always positive.
Thus, the function $$\ln\left(e^{x}\right)$$ is one-one and onto in (R to R).