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I have to draw (by hand obviously) the plot of the following function:

$$f(x)= 13\ln(\frac{x}{|x+1|})-12\ln (x+x^2) +kx,$$

for $k \in \mathbb{R}$. To do so, I have to study the first and second derivative, limits at infinity, and so on. Normally, I do these exercises with functions without parameters. Could you show me how to proceed in this case?

Also, I have to determine for which values of $k$ this function is *bijective*. However, I don’t have a clue about how to proceed, as I’ve never been shown exercises of this kind.

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Due to the first term, the function is only defined for $x>0$, because $\frac{x}{|x+1|}>0$ if and only if $x>0$. So we’ll assume $f\colon(0,\infty)\to\mathbb{R}$.

So first of all rewrite it as

$$

f(x)=13\ln x-13\ln(x+1)-13\ln x-12\ln(1+x)+kx=\ln x-25\ln(x+1)+kx

$$

We easily have

$$

\lim_{x\to0}f(x)=-\infty,

$$

In order that the function is onto $\mathbb{R}$, we need that $\lim_{x\to\infty}f(x)=\infty$. Now

$$

f(x)=\ln\frac{x}{(x+1)^{25}}+kx

$$

and the limit at $\infty$ of the first term is $-\infty$. So this forces $k>0$, because for $k\le0$ we have $\lim_{x\to-\infty}f(x)=-\infty$. When $k>0$ we indeed have $\lim_{x\to\infty}f(x)=\infty$ (verify it).

Now we need that the function is monotonic, so we look at the derivative

$$

f'(x)=\frac{1}{x}+\frac{25}{1+x}+k

$$

Under the hypothesis that $k>0$ this is everywhere positive, so the function is indeed increasing.

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