Are $e^{\sqrt x}$ and $e^{x^2 +1}$ examples of exponential functions?

Question

In many textbooks and online reference the I've seen exponential functions always have a linear variable as a power of constant base and an increasing monotonic graph. Are these $e^{√x}$ and $e^{x^2 +1}$ functions also exponential functions. What is the unique characteristic for which a function is said exponential and do these functions has the same traits. Why they do not have a monotonic graph? Which $e^x$ has. Do they need to satisfy the pre defining properties an exponential function should satisfy i.e. ( base>0, base must not be 1, and power always belongs to real set (R)). Please clarify over each point. Thank you.

Answer 1

Refer to the definition of exponential function, for example from Wikipedia :

The functions such as $f(x)=e^{\sqrt x}$ and $e^{x^2 +1}$ are composite functions on the form $f(x)=e^{g(x)}$. They are not exponential functions according to the above definition, except if $g(x)=cx+d$ with constant $c,d$ .

They are "composite exponential functions" (I am not sure of the correctness of the translation. Please correct it if necessary).

For example in case of $g(x)=\ln(x)$ the fonction $f(x)=e^{g(x)}=x$ is not an exponential function.