There is a fallacy in mathematics, and that is that (3,8) can stand either
for
(1) the point with x-coordinate 3 and y-coordinate 8.
or
(2) the open interval {x|3 < x < 8}
----------------o==============o--------->
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11
Try not to get these confused.
Draw all three lines y=2x+5, y=-3, and y=5x
The red line is y=2x+5. Chop it off on the left
at (-3,-1), and put a closed circle on the left end
and an open circle on the right end at (0,5)
The green line is y=-3. Chop it off both on the left
and right at (-3,0), and put a closed circle at
(-3,0). There will be just one single green point
at (-3,0), because the green line is chopped off on
both sides of that one green point.
The blue line is y=-5x. Chop it off on the left
at (0,0), and put a open circle on the left end
and let it extend indefinitely on the right.
A. Find domain and range.
Look at the graph above:
The left most point is (-3,-1), so x starts the domain at -3.
All the x values to the right of -3 have a y-value, so the
domain is [-3,∞ )
The highest point is (0,5), but it does not include that point,
but it includes all lower values of y, so the range is (-∞,5).
B. Locate intercepts.
It has only one intercept, an x-intercept on the left part y=x+5
Substitute y=0 and get 0=2x+5, or 2x=-5, or x=-5/2=-2.5. So the only
intercept is (-2.5,0). [Notice that both endpoints on the y-axis
are not included. So there is not another x-intercept and no
y-intercept at all.
C. Is f continuous on its domain?
No because the pieces do not connect. To be continuous on the domain
each part must begin at the same point where the preceding part ends.
Edwin
(