We started with the formal definition of a function: commit the
"arrow diagram for f" to memory, including the fundamental
asymmetry (there is a flow to a function, from the domain and its
independent variable to the range and its dependent variable).
We discussed graphical and algebraic means for identifying symmetry in
functions (even and odd functions). I emphasized the importance of symmetry,
and asked you to be on the lookout for it.
In groups you drew some of the wildest, simplest and most useful functions
you could think of. Some observations:
Your wild functions featured:
an exponential function (which grows wildly!)
complex mixes of algebraic formulas that look ugly
lots of wiggles
breaks (aka discontinuities)
a smudge function - very complex!;)
Your simple functions included:
Constant functions (such as f(x)=2)
Linear functions (i.e. f(x)=mx+b), which are in fact the most
important functions we'll use in this class;
a function with an empty domain! Now that's a simple function!
a function with a single point for a domain. It's graph was {(2,5)} - very
simple!
Your useful functions included:
a function illustrating that there's a relationship between the mpg and mph
the compound interest formula (describing how your money grows in the bank)
a parabola (focuses light from stars, from the beams of your car's
headlights and your flashlight, etc.).
It was interesting to note that almost all of your functions involved the real
numbers. In fact, most of the functions we study in calculus will have domains
and ranges of real numbers.
Nice job!
We looked at an example of a problem in which we computed domain and range
in order to plot it using our calculators in the most appropriate window.
We found the domain by using some facts about the shape of the quadratic
function. It's important to know the shapes of various functions, especially
when we turn to our next topic: modeling.
Notes:
There's an appendix on invervals and solving inequalities in our text (Appendix A)