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Find an expression for the function whose graph is the given curve.

$f(x)=\left\{\begin{array}{ll}{-x+3} & {\text { if } 0 \leq x \leq 3} \\ {2 x-6} & {\text { if } 3<x \leq 5}\end{array}\right.$

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Missouri State University

Oregon State University

Harvey Mudd College

Boston College

Okay, We're given the graph of peace vice function, and we want to express the function algebraic Lee. And so we need to find the equation of each piece. And each piece is a line, a piece of a line. So to find the equation of a line, you need two points. So notice that we have for the left most line segment, we have the 20.3 and we have the 0.30 Let's use those points to find the slope and the equation of the line. So for the slope, we have my two minus y 13 minus zero over x two minus x 10 minus three. And we end up with three over negative three, which is negative one. And for the Y intercept noticed that you can tell it's just the 0.3 So be is three. So we have the line. Why equals negative one X plus three. So in our piece wise definition, that's going to be one of the pieces. So the y equals is the f of X equals And so we have negative X or the opposite of X plus three. Now what's the domain from that piece. Well, it's going from X equals zero to X equals three. So we can feel that in a swell zero is less than or equal to X is less than or equal to three. Now let's figure out the equation of the other line. So we have the 0.30 and we also have the 0.54 notice that we have a rise of five and a run of one of with the rise of five and a run of two. Excuse me. So the slope of that segment is five house, and what we can do is use point slope form and use one of our points and use our slope and find the equation of the line. So let's use the 0.30 So we have y minus zero equals five halves, times X minus three. So that would be why equals five halves X minus 15 house. Okay, I heard some of you telling me to back up and change my rise to a four. So let's do that. The rises for so about that. So the slope is actually to So we have two X minus six. There. That's better. Okay, So we put that piece into are piece wise function that why equals part is already there. That's the f of X equals part. So we just write the two X minus six and then what's the domain for that piece? Well, that piece went from X equals 32 X equals five So we can say three is less than X is less than or equal to five. Notice that I already put the equal to part on the three on the previous peace. So I don't use it again on the new piece, we only have equals two on one piece.