H: Domains and ranges for a zoo of functions using interval notation.
The Magnitude or Absolute Value Function:
Introductory Exercise (4): Join the adjacent points in the list
(-4,4), (0,0), (4, 4) by straight line segments. The resulting graph should
satisfy the vertical line rule. Use the graph to form a table of values
for the the resulting function f at integer points in the interval [-4, 4].
Find, guess if you have to, a formula for f(x) when x > 0 and another
formula for f(x) when x < 0.
The foregoing may allow students to understand the piecewise definition of
the absolute value functions, and alternatives to it.
| f(x) = |x| = { |
x if x > 0
-x if x < 0 |
Here range(f) = [0, +oo[ = set non-negative real numbers, and domain(f) =
IR = set of real numbers.
Alternative Method for calculating absolute value: Each real number is
given by a magnitude and + or - sign prefix. The absolute value |x| of a
number x is simply its magnitude as is or with a + sign prefix (as you like)
Applications of the Absolute Value Function- See rectangular
coordinates and associated distance
formulas for line and plane.
Related Material: Distance
and midpoint mormulas See too Chinese Square Proof of the Pythagorean
theorem online.
Step Function g(x) = [x] = greatest integer < x
Here range(f) = Z = set of integers; and
domain(g) = IR = set of real numbers
Saw Tooth Function Saw(x) = x - [x] = remainder on division by
1.
Here domain(g) = IR = set of real numbers while range(f) = [0,1[ =
the half-open interval that includes its left end point 0 and excludes its
right-end point 1 = [0, 1) in outside of Quebec notation.
The further introduction of exponents, radicals, polynomials, ratonal
functions provides further opportunities to define functions, to describe when
they can be computed (domains) and their values (ranges).
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