Section+1.2

Homework for Section: Day 1- pg. 102 #1-19 odd Day 2- pg 102 #21-39 odd Day 3- pg102 # 41-53 odd  **1.2. Functions and Their Properties**
 * Chapter 1: Functions and Graphs ** pg. 86-105

Function: a function from set D to R is a rule that is assigned to every element in D and a unique element in set R. **Domain**: the input values (x) // independent // **Range**: the output values (y) // dependent // Example: //y=2x+7// x --> 2x+7 --> y 3 --> 2(3)+7 --> 13   Function Notation: y=f(x) à  “f of x”   f= the name of the function  x= the input variable  y= output variable <span style="COLOR: rgb(255,153,0)"> <span style="COLOR: rgb(255,153,0)"> <span style="COLOR: rgb(255,153,0)"> <span style="COLOR: rgb(92,148,87)">These are some different ways to see if an equation is a function. <span style="COLOR: rgb(69,104,74)">__**Logic**:__ a function can only have one //unique// output. So if an equation has //two// possible solutions, it is //not// a function.
 * Function Definition and Notation**
 * //<span style="FONT-SIZE: 14pt; COLOR: rgb(51,153,102)"> Is it a Function?? //**

__<span style="COLOR: rgb(51,153,102)"> **<span style="COLOR: rgb(75,124,83)">Graphically ** __ <span style="COLOR: rgb(51,153,102)"> <span style="COLOR: rgb(51,153,102)"> Vertical Line Test: For every vertical line, if any line touches the graph at more than one point, then it is n<span style="COLOR: rgb(0,0,0)">o t a function. <span style="COLOR: rgb(146,17,17)"> <span style="COLOR: rgb(67,10,113)"><span style="COLOR: rgb(187,22,26)"><span style="COLOR: rgb(146,17,17)"><span style="COLOR: rgb(188,6,6)">** Domain and Range ** <span style="COLOR: rgb(67,10,113)"><span style="COLOR: rgb(146,17,17)"><span style="COLOR: rgb(187,22,26)"><span style="COLOR: rgb(188,6,6)">Domain: Think- "What values can I //input// into the equation?"

//Example: f(x)=2.-7 Domain = all real numbers

Range: Think- "What values can i get out of this equation?"// <span style="COLOR: rgb(20,4,103)"> **Continuity** __Continuous at all x__

This graph is continuous everywhere. It has no breaks. //Example: Try graphing g(x)= (x=3)(x-2)//

__Removeable discontinuity__ These graphs can be patched by redefining f(a) so as to plug the hole in the continuous line. //Example: Try graphing h(x)= x^2-4/x-2//

<span style="COLOR: rgb(20,4,103)">__Jump discontinuity

__This is a discontinuity that is not fixable. There is more than just a hole at x=a. There is a jump in function values that makes the gap impossible to fix with a single point no matter how we try to redefine f(a).

__Infinite Discontinuity

__This is a function with infinite discontinuity at x=a. The discontinuity is not removeable. //Example: Try graphing f(x)= x+3/x-2//

<span style="FONT-SIZE: 110%; COLOR: rgb(213,16,16)"><span style="COLOR: rgb(212,100,28)">**<span style="COLOR: rgb(212,103,8)">Increasing and Decreasing Functions **   <span style="COLOR: rgb(212,100,28)"><span style="COLOR: rgb(213,16,16)"><span style="COLOR: rgb(212,103,8)"> >
 * <span style="COLOR: rgb(212,103,8)">a function f is **//increasing//** on an interval if, for any two points in the interval, a positive change in x results in a **positive change** in f(x).
 * <span style="COLOR: rgb(212,103,8)">a function is **//decreasing//** on an interval if, for any two points in the interval, a positive change in x results in a **negative change** in f(x)
 * <span style="COLOR: rgb(212,103,8)">a function f is **//constant//** on an interval if, for any two points in the interval, a positive change in x results in zero change in f(x)

<span style="COLOR: rgb(48,90,65)"> **Bounded or Unbounded?**

1. A function is bounded below if there is a number less than or equal to every numer in the range of the function. This number would be called the **lower bound.** 2. A function is bounded above if there is some number that is greater than or equal to every number in the range of the function. This number would then be called the **upper bound**. 3. A function is bound if is is bounded both above and below.

<span style="COLOR: rgb(95,49,135)"> **Local and Absolute Extrema**


 * Local**- The maximum or minimum range value on some tiny interval.
 * Absolute**- The maximum or minimum range value of the entire function.

//Example: Graph **f(x)=x^4-7x^2+6x** on your calculator. You can find the extremas by using the 'minimum' and 'maximum' functions under the 'calculate'. There are two local minimum values and one local maximum value on this graph. We refer to the min/max's as the y coordinates of the range values.

Solutions: y=-1.8 (x= 1.6) - minimum y= 1.32 (x=.46) - maximum y=24.1 (x=2.06) - minimum//

** The graphical sense of the word "symmetry" in math has the same meaning as it does in art. It looks the same on both sides, or looks the same when viewed in more than one way. There are thee particular types of symmetry.
 * Symmetry

1. "**Even symmetry**" occurs over the y axis. It looks the same on the left side of the y-ais as it does on the right.

//Example: f(x)=x^2// 2. **Odd Symmetry**- In these functions, every given point (a,b) matches up with a point (-a,-b). To determine odd symmetry, you whould replace the variable with its opposite. If it has odd symmetry, it should stay the same when simplified.

//Example: y=x^3// 3. **Over x-axis** (not a function!!!!!) These graphs look the same above the x-axis as it does below. However, graphs with this type of summetry are not functions (Notice that the vertical line test doesn't work). Still, we can say that (x,-y) is on the graph whenever (x,y) is on the graph.

//Example: x=y^2//