Fibonacci is famous for his contributions to number theory.
In his book, Liber abaci he introduce the Hindu-Arabic place-valued decimal systems and the use of Arabic Numerals into Europe.He introduced the bar we use in fractions, previous to this, the numerator has quotations around it. The square root notation is also is Fibonacci method.
An inequality is a relation that makes a non-equal comparison between two numbers or other mathematical expressions. It’s mostly used to compare 2 numbers on the number line by sizes.
B. If Karl runs 1 mi, then he runs 5280 ft.
If A then B and if B then C, then A-->C
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.
--------------------------------------------------
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
--------------------------------------------------
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".