Mr. Dreyer

Algebra 2 Lv 1

Fri Sept 5

1Warmup

1. Check seating chart
2. Welcome/expectations signatures on desk
3. Got calculators?

4. Consider the function defined by this two-part rule:

   ${\displaystyle f\left(x\right)=\{\begin{array}{ll}0& \mathrm{if}x\mathrm{is}\mathrm{even}\\ 1& \mathrm{if}x\mathrm{is}\mathrm{odd}\end{array}.}$

   Evaluate:

   1. $f\left(2\right)$
   2. $f\left(153\right)$
   3. $f\left(0\right)$
5. If we know that $f\left(x\right)=1$, do we know the value of $x$?

2G Block: Textbooks

• Go in alphabetical order
• Find your book (see spreadsheet)
• Write your name & mine (Dreyer) in bookplate immediately

3Classwork/homework head start

Section 2.1, page 71: 9, 10, 15, 16, 24, 25, 27, 33, 42, 50, 52

4Diagnostic test

• Mon Sept 8
• Not “high stakes”
• Very high or very low scores may be considered for class changes

5Functions may be thought of as:

• A rule assigning output values to input values
• A rule in the form of an equation, e.g. $f\left(x\right)=x^2-1$ or $y=x^2-1$
• A set of ordered pairs in which the first element of any pair is not equal to the first element in any other, sometimes shown as a table. If the function is also expressed as an equation, the ordered pairs represent all the solutions to the equation
• A graph, which is just a picture of a set of ordered pairs (what are the ordered pairs?) that obeys the vertical line test.
• What does the vertical line test have to do with the “distinct first elements” test of the ordered pairs representation above?

6Functions you already know, but may not think of as functions

• The “double it” function:

  ${\displaystyle f\left(x\right)=2x}$

  $x$	$f\left(x\right)$
  1
  	4

• The “negative” function:

  ${\displaystyle \begin{array}{rcl}f\left(x\right)=& -x& \\ (\mathrm{or},\mathrm{if}\mathrm{you}\mathrm{prefer})f\left(x\right)=& -1x& \end{array}}$

  $x$	$f\left(x\right)$
  2	$-2$
  1
  	0

• Absolute value function:

  ${\displaystyle f\left(x\right)=|x|}$

  which can also be defined as

  ${\displaystyle f\left(x\right)=\{\begin{array}{ll}x& \mathrm{if}x⩾0\\ -x& \mathrm{if}x<0\end{array}.}$

  $x$	$f\left(x\right)$
  $-2$	2
  3	3
  	1

• Even operations like addition can be thought of as functions with two inputs:

  ${\displaystyle f(x_1,x_2)=x_1+x_2}$

  $x_1$	$x_2$	$f(x_1,x_2)$
  1	2	3
  $-4$		$-1$
  	7	7

  We won't otherwise deal with functions of two inputs for awhile, though.

• Here's a word description of a function: Square the input number and subtract one.

  What is the equivalent equation?

  ${\displaystyle f\left(x\right)=x^2-1}$

  Here is a table to describe a bit of the function:

  $x$	$f\left(x\right)$
  2	3
  1
  	8

7Some buzzwords

• Domain: set of legal input ($x$) values
• Range: set of legal output ($y$ or $f\left(x\right)$) values
• Independent variable: the input variable
• Dependent variable: the output variable (think: it depends on the independent variable)
• Linear functions are functions that can be written $f\left(x\right)=mx+b$ for some numbers $m$ and $b$. What do the graphs of linear functions look like? Is the absolute value function $f\left(x\right)=|x|$ a linear function?

8Relations

• A relation is simply a set of ordered pairs. Unlike a function, the first elements do not have to be distinct

  • Are all functions relations? Are all relations functions?
• What's the difference between the graphs of a function and a relation?
• Is a circle a graph of a function? A relation?
• What equation would generate a graph of a circle centered on $(0,0)$ of radius 1? How does this equation differ from an equation that defines a function?
• Does the concept of domain apply to a relation? How about range? Independent variable? Dependent variable?