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See the attached image for the graph. The piecewise function is simply a collection of functions glued together to form a sort of "frankenstein function" of sorts. Here's how it works. If x is between 0 and 1 (including 0 but excluding 1), then f(x) = -2 for all x values mentioned. This is exactly what the first row of the piecewise function says. Then the second line says "if x is between 1 and 2, including both endpoints, then f(x) = 3-x". Finally the last line says f(x) = 1 as long as x is larger than 1 but smaller than 4. The value x = 4 is allowed for this piece but x = 1 is not allowed.
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Now onto function evaluation. We want to evaluate f(1) so we need to find f(x) when x = 1. If x = 1, then we'll use the second piece because the first piece excludes x = 1. So
f(x) = 3-x
f(1) = 3-1
f(1) = 2
is the first answer for the function evaluation section.
The value of f(3) is equal to 1 since we resort to using the third piece of the piecewise function (x = 3 is in the range from x = 2 to x = 4)
The value of f(-5) is undefined simply because we're out of the range defined by the piecewise function. The smallest x can get is x = 0.
Finally, f(2) is equal to 1 since...
f(x) = 3-x
f(2) = 3-2
f(2) = 1
again we use the second piece of the piecewise function because x = 2 is in the range 1 <= x <= 2
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In summary:
f(1) = 2
f(3) = 1
f(-5) = undefined
f(2) = 1
Once again have a look at the graph to see how the points match up. I've color-coded the graph so that each piece is a different color. The first piece is in red, the second in blue, the third in green. Note the open hole at point B to indicate "don't include this point as part of the graph".
The area is the sum of 'n' rectangle areas.
The width of the rectangle is size of interval, (domain size)/n
The height of each rectangle is f(interval) as interval moves along domain.
For this example, domain size = 3-1 = 2
size of interval = 2/n
height varies from f(1) to f(3), increasing by 2/n each time.
f(1+(2/n)i)
Putting this together, the area is the sum of:
Since you are given the function f(x). Sub input into f(x) to get area in terms of n and i.
Finally, the summation is:
The width of the rectangle is size of interval, (domain size)/n
The height of each rectangle is f(interval) as interval moves along domain.
For this example, domain size = 3-1 = 2
size of interval = 2/n
height varies from f(1) to f(3), increasing by 2/n each time.
f(1+(2/n)i)
Putting this together, the area is the sum of:
Since you are given the function f(x). Sub input into f(x) to get area in terms of n and i.
Finally, the summation is:
