Question 1092395: graph of function
y= (|x-3|+|x+1|)/(|x+3|+|x-1|)
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The behavior of the function changes each time we pass a point where one of the expressions in absolute value symbols is equal to 0.
For checking the graph we come up with, it is useful to evaluate the function at each of those points.
So the places where the behavior of the graph changes are (-3,2), (-1,1), (1,1), and (3,.5)
(1) For x < -3...
This is a segment of a branch of a hyperbola with horizontal asymptote y=1 and vertical asymptote at x = -1.
(2) For -3 < x < -1...
This is of course a segment of a line.
(3) For -1 < x < 1...
And this is clearly a constant function.
(4) For 1 < x < 3...
This is a segment of a branch of a hyperbola with horizontal asymptote y=0 and vertical asymptote at x = -1.
(5) And for x > 3...
And this is part of the other branch of the same hyperbola as in (1).
Here is a graph of the function. Sorry; I haven't yet figured out how to draw the graph of a piecewise function.
(1) red line from (-infinity,1) to (-3,2)
(2) green line from (-3,2) to (-1,1)
(3) blue line from (-1,1) to (1,1)
(4) purple line from (1,1) to (3,.5)
(5) red line again from (3,.5) to (infinity, 1)
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