Question

In Problems 13–32, use the accompanying graph of $y=f\left(x\right)$.

What is the domain of $f$?

Short answer

The domain is $$\begin{gathered} x\in [-8,-6)\cup (-6,4)\cup (4,6], \\ or, \\ x\in [-8,6]-\{-6,4\}. \end{gathered}$$.

Step-by-step solution

Step. 1 Given

Domain is the value of x on which the function is defined. We are given with a graph so in graph domain will be the set of all points on x-axis for which the function is defined.

Step. 2 Domain calculation

Since we can see clearly that the graph is defined over $x\in [-8,6]$but hold on,

we have some critical points on which we have to check the value of f(x) is valid or not.

these are, $x=-6,-4,2,4$.

Firstly, at $x=-6$the graph goes to $\infty$so it must not be in the domain of $f\left(x\right)$.

Secondly, at $x=-4$the value of $f\left(x\right)$is 2 so it will be inclusive in the domain.

Next, $x=2$here also the value of $f\left(x\right)$is 3 so it will also be inclusive in the domain.

Finally, $x=4$here we can clearly see that it is open so it will not be the part of domain.

Step. 3 Final gist of domains

From step 2 we conclude that

$x\in \left\{\right(-8,-6)\cup (-6,-4)\cup (-4,2)\cup (2,4)\cup (4,6\left)\right\},$

and we see that we have to include $x=-8,-4,2,6$these in the domain so final domain will be,

$$\begin{gathered} x\in [-8,-6)\cup (-6,4)\cup (4,6], \\ or, \\ x\in [-8,6]-\{-6,4\} \end{gathered}$$