Use the provided information to sketch the graph of $f$. Assume that $f$ is continuous on its domain and that all intercepts are included in the table of values.

Domain: All real $x$, except $x=-1;$

$f(-3)=2,f(-2)=3,f(0)=-1,f(1)=0;$

$f'(x)>0\ \text{on}\ (-\infty,-1)\ \text{and}\ (-1,\infty)$

$f''(x)>0\ \text{on}\ (-\infty,-1); f''(x)<0\ \text{on}\ (-1,\infty);$

vertical asymptote: $x=-1;$

horizontal asymptote: $y=1$

Solution

VerifiedTo sketch the graph of $f$, we need to determine the intervals where it is increasing, decreasing, concave upward, concave downward, and the asymptotes.

Note that if $f'(x)$ is positive then $f(x)$ is increasing, and if $f'(x)$ is negative then $f(x)$ is decreasing.

$f'(x)$ is positive on $(-\infty,-1)$. So, $f(x)$ is increasing on $(-\infty,-2)$.

$f'(x)$ is positive on $(-1,\infty)$. so, $f(x)$ is increasing on $(-1,\infty)$.