Question

1. sketch the graph of a function f that satisfies all of the given conditions. $lim_{x \to 0} f(x)=1$, $lim_{x \to 3^{-}} f(x)=-2$, $lim_{x \to 3^{+}} f(x)=2$, $f(0)=-1$, $f(3)=1$.

Explanation:

Step1: Mark known points

Mark the points $(0, - 1)$ and $(3,1)$ on the graph since $f(0)=-1$ and $f(3) = 1$.

Step2: Consider limits at $x = 0$

As $\lim_{x
ightarrow0}f(x)=1$, the function approaches $y = 1$ as $x$ approaches $0$. But $f(0)=-1$, so there is a hole at $(0,1)$ and a point at $(0,-1)$.

Step3: Consider limits at $x = 3$

Since $\lim_{x
ightarrow3^{-}}f(x)=-2$, the function approaches $y=-2$ as $x$ approaches $3$ from the left. And $\lim_{x
ightarrow3^{+}}f(x)=2$, the function approaches $y = 2$ as $x$ approaches $3$ from the right. Also $f(3)=1$, so there are holes at $(3,-2)$ and $(3,2)$ and a point at $(3,1)$.

Step4: Sketch the curve

Connect the parts of the function in a smooth - enough way (not unique) such that the above - mentioned limit and function - value conditions are met.

Answer:

A hand - sketched graph with points $(0,-1)$ and $(3,1)$ marked, holes at $(0,1)$, $(3,-2)$ and $(3,2)$ and the curve approaching the appropriate limit values as $x$ approaches $0$ and $3$ from the relevant directions.