함수의 증가와 감소

The domain of $f(x)$ is all real numbers except $-3$ (because when $x=-3$ the denominator is zero). For $x\neq -3$, we differentiate using the quotient rule: $$f'(x)=\frac{e^x(x+3) - e^x(1)}{(x+3)^2} = \frac{e^x}{(x+3)^2} (x+2)$$ Since $e^x$ and $(x+3)^2$ are positive for every $x$ in the domain of $f(x)$, the sign of $f'(x)$ is the same as the sign of $x+2$. We conclude that $f(x)$ is increasing for every $x$ in its domain with $x+2>0$. That is, over the open interval $(-2,\infty)$.

Since we can't take the square root of a negative number, $f(x)$ is only defined when $x \ge 1$. Furthermore, since we can't have zero as a denominator, $x=-2$ is not in the domain — but as long as $x \ge 1$, we also have $x \ne -2$. So, the domain of the function is $[1,\infty)$.

In order to find where $f(x)$ is increasing, we find where $f'(x)$ is positive. $$f'(x)=\frac{\frac{2x+4}{2\sqrt{x-1}}-2\sqrt{x-1}}{(2x+4)^2} =\frac{(x+2)-2(x-1)}{\sqrt{x-1}(2x+4)^2} =\frac{-x+4}{\sqrt{x-1}(2x+4)^2}$$ The denominator is never negative, so $f(x)$ is increasing when the numerator of $f'(x)$ is positive, i.e. when $4-x>0$, or $x<4$. Recalling that the domain of definition for $f(x)$ is $[1,+\infty)$, we conclude that $f(x)$ is increasing on the open interval $(1,4)$.

The domain of arctangent is all real numbers. The domain of the logarithm function is all positive numbers, and $1+x^2$ is positive for all $x$. So, the domain of $f(x)$ is all real numbers.

In order to find where $f(x)$ is increasing, we find where $f'(x)$ is positive. $$f'(x)= \frac 2 {1+x^2} - \frac{2x}{1+x^2} = \frac{2-2x}{1+x^2}$$ Since the denominator is always positive, $f(x)$ is increasing when when $2-2x>0$. We conclude that $f(x)$ is increasing on the open interval $(-\infty,1)$.