Question

Confused on how to do these Piecewise-Defined evaluations..

Answer 1

check the picture below.

notice the dashed lines, those are the "continuations" lines for the subfunction of the piecewise, however, the subfunctions have a range each, so that part is not part of the piece-wise function.

anyway, notice, the solid dots on the ends mean, that endpoint number is included, and the "holes" in the ends, mean that endpoint is not included, or not part of the range, something like 2 < x, simply means, "x" is greater than 2, but not 2 itself, so 2 is not a value for "x", just greater numbers.

so, you can see there what f(1), f(3) and f(2) are, hmmm what happened to f(-5)?  well, is out of range, is not part of the piece-wise, so N/A (not available).

Answer 2

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".