\((x+1)(x+2) = x(x+2) + 1(x+2)\) If you are unsure, typically easy enough to plug numbers in: \((2+1)(2+2) = 3\cdot 4 = 12\)

\(\frac{73}{92} = \frac{1+72}{20+72} = \frac{1}{20}\)

\(\frac{36}{9} = \frac{9\cdot 4}{9 \cdot 1} = \frac{4}{1} = 4\)

WHY ARE THEY WRONG

Definition: “A rule for a relationship between an input, or independent, quantity and an output, or dependent, quantity in which each input value uniquely determines one output value. ”The output is a function of the input“

Good function: How much money you make for working x many hours Is this a good function?: capsule vending machines input a quarter output is random. I think thinking is often constricted, but functions are a truely abstract notion.

Notation: \(f(x) = 2x + 3\) \(f(1)\) is now a number. It is no longer a function.

Some functions:

1. \(f(x) = \sqrt{x} = x^{1/2}\)
2. \(f(x) = \sqrt{3}{x} = x^{1/3}\)

Output is not always defined:

1. \(f(x) = \sqrt{x}\)
2. \(f(x) = \frac{1}{x}\)

One-to-one functions: “Every output has exactly one input” It is always that one input has exactly one output. Which one is one-to-one

1. \(f(x) = x^2\)
2. \(f(x) = 2x\)

How to graph functions:

BOOK:

1. Examples and Definition of function
   1. Ex: How much money you are owed after working \(x\) hours
   2. Ex: Given the month, how many days are in it?
   3. Nex: Given a number of days what month is it?
   4. Nex: Random output like a capsule vending machine.
2. 1-1 function
3. Function notation
4. Tables
5. Solving and evaluating functions
6. Graphs as functions
   1. Vertical line test: Shows that one input has one output
   2. Horizontal line test: Shows if a function is one-to-one
7. Formulas as functions