I have a graph of a piecewise function below, and I am having trouble figuring out the domain of the function in interval notation.
My answers are:
Domain: $[-7, -1)\cup(-1, \infty)$
Range: $[-6, \infty)$
I am told my range is correct but my domain is wrong, and I can't seem to figure out why.
$\endgroup$ 14 Answers
$\begingroup$Well, let's look at your graph, and where the discontinuities lie. The first is at $x=-5$, a removable discontinuity, the second at $x=-1$, an asymptotic discontinuity, and the third at $x=3$, a jump discontinuity.
Now we notice that the function simply isn't defined at $x=-5$, so that can't be part of the domain. The same can be said for $x=-1$, but the function is defined for $x=3$, so it's part of the domain.
Hence, our answer is $$[-7,\infty)\setminus\{-5,-1\}$$or$$[-7,-5)\cup(-5,-1)\cup(-1,\infty)$$
$\endgroup$ 2 $\begingroup$Maybe that's due to the difference between a blue dot and a white dot. I guess the white dot means that the function is not defined there. Hence the domain would exclude $-5$.
$\endgroup$ $\begingroup$The domain is
$$[-7,-5)\cup (-5,-1)\cup (-1,8]$$
the range is $$[-6,+\infty)$$
$\endgroup$ $\begingroup$When you see the white dot, it just means that the function is not defined at that point. So, $f(-5)$ is undefined, also $f(-1)$ is undefined, but $f(3) = 0$.
Also, the domain does not stop at $x = 8$; it goes all the way to $\infty$. Some textbooks add an arrow to show it goes to $\infty$. Finally, your domain is: $$[−7,−5)∪(−5,−1)∪(−1,\infty).$$
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