Key Terms
Relation
Any set of ordered pairs. Domain: the set of all input values (x-values; independent variable).
Function
A relation where each input leads to exactly ONE output. No x-value repeats with a different y-value.
Example (fails)
Input 5 maps to both 40 and 42. Same input, two outputs.
Example
F(x) = x^3 - 12x Increasing on (-inf, -2) U (2, inf). Decreasing on (-2, 2).
Evaluating
You know the input; find the output.
Solving
You know the output; find the input(s).
To solve from a table
Scan the output column for the target value. Every row that matches is a valid solution.
To evaluate f(a)
Find x = a on the graph; read the y-coordinate. To solve f(x) = b: find y = b on the graph; read every x-coordinate that
Regular function
Each input has exactly one output. One-to-one function: each input has exactly one output AND each output has exactly on
Logic
A horizontal line at y = b shows all inputs that produce that output. More than one hit means one output comes from mult
Domain
All valid inputs - what you're allowed to plug in. Range: all possible outputs - what comes out.
Two things can restrict the domain
1. Denominators: you cannot divide by zero.
Standard notation
F(x) = { formula 1, if x is in region 1 { formula 2, if x is in region 2 { formula 3, if x is in region 3
Example - data plan pricing
C(g) = 25 if 0 < g < 2 C(g) = 25 + 10(g-2) if g >= 2
Formula
Average rate of change = [f(x2) - f(x1)] / (x2 - x1) = delta-y / delta-x