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Problem 41 Hard Difficulty

Evaluate $ f(-3) $ , $ f(0) $ and $ f(2) $ for the piecewise defined function. Then sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
x + 2 & \mbox{if $ x < 0 $}\\
1 - x & \mbox{if $ x \ge 0 $}
\end{array} \right.$

Answer

$f(x)=\left\{\begin{array}{ll}x+2 & \text { if } x<0 \\ 1-x & \text { if } x \geq 0\end{array}\right.$
$f(-3)=-3+2=-1, f(0)=1-0=1,$ and $f(2)=1-2=-1$

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Video Transcript

here we have a piece vice function, and we're going to find some function values and then sketch the graph. So first we want to find f of negative three. And so we figure out which piece to plug it into notice. That one piece has a domain of X is less than zero. So negative three would fall into that category, so we would have negative three plus two, and that's negative one. Now let's find F of zero. So we have to figure out which piece to plug zero into notice that we have a piece where the domain is X is greater than or equal to zero. So zero falls into that group, so we would use one minus X, and we would have one minus zero. And that's one. Now we'll find F of to So which piece, should that fall into? Well, two is in. The X is greater than or equal to zero category, so we have one minus two, and that's negative one. So when we graph are when we sketch our graph, these points should turn out to be points on a graph. Now let me demonstrate to you the way I like to graph a piece. Wise function. First of all, I'd like to stop and think about what I'm expecting to see. So f of X equals X Plus two should look like a line. A part of a line and f of X equals one minus X should also look like a part of a line. Now the line stop at zero because that's where we have our break from one piece to another. So here's what I like to dio. I like to find a couple points, including the endpoint. So on the first piece, the endpoint is zero. It's not actually going to be a point. It's going to be an open circle because we're not including zero. So we're gonna have an open circle at the 00.2 Then we want another point that that line will go through what we just so happen to have found one already. We found the point negative three negative one. So what we're going to do to draw that peace is start at the 0.2 with an open circle and then go through the point negative three negative one and keep going. That's that piece of the graph for the other piece. The end point is zero. So we're going to start at 01 because we're including zero here. That's going to be a closed circle. And then we need a second point that the graph goes through. We just so happen to have already found the point to negative one. So we started 01 with a closed circle. We go through the point to negative one and we keep going and there's a graph.