Proving convexity via monotonicity
Consider the function , with values
.
We can express the function as the composition of the function
with the function with values
That is, . Since belongs to the domain of for every , the domain of is indeed the whole real line.
The functions are both convex, and is monotone increasing (note that the domain of is chosen to ensure monotonicity). Hence, by the monotonicity property, the composition is also convex.
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