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Sketch the graph of the function.

$ f(x) = \left\{

\begin{array}{ll}

\mid x \mid & \mbox{ if $ | x | \le 1 $}\\

1 & \mbox{ if $ | x | > 1 $}

\end{array} \right.$

To graph $f(x)=$ $\left\{\begin{array}{ll}|x| & \text { if }|x| \leq 1 \\ 1 & \text { if }|x|>1\end{array}\right.$ $\operatorname{graph} y=|x| \text { (Figure } 16)$

for $-1 \leq x \leq 1$ and graph $y=1$ for $x>1$ and for $x<-1$

We could rewrite $f$ as $f(x)=\left\{\begin{array}{ll}1 & \text { if } x<-1 \\ -x & \text { if }-1 \leq x<0 \\ x & \text { if } 0 \leq x \leq 1 \\ 1 & \text { if } x>1\end{array}\right.$

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Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

all right, we're going to graft this piece wise function. So let's start by thinking about what each piece is expected to look like. So we have f of X equals the absolute value of X. And if you think about that function, it typically looks like a V shaped graph with its Vertex at 00 And then we have f of X equals one. So if you think about what Y equals one looks like it would look like a horizontal line at a height of one. So we're going to have a piece of each of those. And the second part of the statement tells us the domain for each piece. When you see the absolute value of X is less than or equal to one, it means the same thing as negative. One is less than or equal to X is less than or equal to one. And when you see that the absolute value of X is greater than one, it means X is greater than one or X is less than negative one. Okay, so that means we're going to have the absolute value graph, but only the part that falls between negative one and one. So that's going to be from this point to this point. We have that V shaped graph. Beyond that, we're going to have the horizontal line, so we're going to have a horizontal line at a height of one to the right and to the left of that be and there's a graph.