Is $f: (0,\infty ) \to \mathbb{R}$ , $f(x):=\ln(x)+e^x$ injective?

Question

i showed that $f$ is strictly monotonically increasing:

$\ln(x+1)+e^{x+1} > \ln (x)+e^x$

Both $\ln$ and $\exp$ are monotonically increasing function, this inequaltiy is obviously true, or do I have to prove that fact?

Answer 1

Take$$\begin{array}{rccc}f\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}x+1&\text{ if }x\in\mathbb Z\\x&\text{ otherwise.}\end{cases}\end{array}$$Then $(\forall x\in\mathbb R):f(x+1)>f(x)$. However, $f$ is not monotonically increasing.

It is more natural to deduce that your function is monotonically increasing from the fact that it is the sum of two monotonically increasing functions.