how can I find the monotonicity of $g(x)=f(e^x-1)-f(2+e^{-x})$ ?

Question

If a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is strictly increasing in $\mathbb{R}$ then how can I find the monotonicity of $$g(x)=f(e^x-1)-f(2+e^{-x})$$ using the definition of monotonicity?

Answer 1

$x \mapsto e^x -1$ is increasing so $x \mapsto f(e^x-1)$ is also increasing (the composition of two increasing functions is an increasing function). $x \mapsto 2 + e^{-x}$ is decreasing so $x \mapsto -f(2 + e^{-x})$ is increasing (the composition of two decreasing functions is an increasing function).

So, $g(x)$ is the sum of two increasing functions.

Answer 2

$f(e^x-1)$ increases and $f(2+e^{-x})$ decreases, which says that $g$ increases.