There are 3 very different skills this week. For functions we study section 1.4. This is on how functions can be combined together. We also step back and review lines in section p3. As well, we begin to learn how to use the calculator.
Functions can be added, multiplied, subtracted and divided very easily. After all they are just numbers once they are evaluated. The only difficult part is notation. For example the function $f(x) + g(x)$ is simply a new function whose value for a given $x$ is simply the sum of the value of two seperate functions. So for example, if $f(x) = x^2$ and $g(x) = \sqrt{x}$ then $h(x) = f(x)+g(x)$ has $h(4) = f(4)+g(4) = 16+2 = 18$.
One new way to combine functions is compositition. For example the output of one function may be put into the input of another. Suppose you have a function which tells you how much you make as a function of hourse worked. And another which tells you how much you get to keep after taxes as a function of how much you make. Then to find your take home pay, you use the first function with the number of hours you worked and then feed this into the second function to get an answer. This is composition.
For example, suppose $f(x) = x^2$ and $g(x)= x+5$ then $f(g(x)) = (g(x))^2 = (x+5)^2$. And $g(f(x)) = f(x) + 5 = x^2 + 5$. Make sure these make sense. What is $f(f(x))$?
The line is one of the basic objects of mathematics. Part of being a student of math is learning about lines. A big part. In this class we do a review of some basics we hope you have already. Here are some highlights.
A line is the graph of an equation that typically looks like
$$y = mx + b \quad\quad \text{point-slope form}$$
The $m$ is the slope and is given by the equation
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
better known as the rise over the run formula. This tells us how fast the line is increasing. If the line is straight up and down the slope is $\infty$, if it is flat it is 0. If it points up the slope is positive, if it points down it is negative.
The $b$ is the intercept on the $y$ axis. Most line will intersect the $y$ axis (which don’t?). It happens just once (except in odd occasions. Again which ones?) and this is $b$.
There are other forms for the line depending on your taste
$y-y_0=m(x-x_0)$
$y=mx+b$
$Ax + By + C = 0$
Part of Math 123 is learning how to use a graphing calculator. If you are already familiar with one from a previous class, you are set. If not, you need to learn.
You will need to have the following skills on a calculator:
entering expressions such as $2^3$, $\sqrt{3}$, fractions, functions such as $sin(12)$ (later). Most students know this already, but you need to remind yourselves of the importance of parentheses.
Graphing functions and reading graphs. The graphing calculator can easily graph a function, but you need to learn how to do
Enter in a function of a variable
Enter a domain for the x-values graphed (an xmin and xmax)
Enter in the range of the y-values to show (a ymin and ymax)
Plot the graph.
All of these steps are covered in the math department’s notes and in the TI-82/83 tutorial that is available on line. (See Web Work in the weekly assignment)
Make a table of values and investigate numbers this way. Essentially, you will learn how to make a table of $x$ values, then make the corresponding $y$ values.
There are numerous other features that these calculators can do. For example, find minmum and maxiumum values, find intersetions with the axes or other functions, etc. You may find these useful. We can discuss how to do these things in the discussion board.