Functions and Absolute Values

In class we talked about how absolute value and functions can be related and/or can be found in the same equation and we learned how to graph absolute values and how to graph inequalities with absolute values. 

Functions and Absolute Value:

-Functions are a specific type of relation because they have only one input value for each output value 

-Absolute Value is the distance a number is away from 0 on the number line

• When using absolute value the number is always in "| |" 
• Inside the brackets (| |) the number CAN be NEGATIVE but because it is the amount away from 0 the absolute value of a number is ALWAYS POSITIVE.

Ex. |5|=5 and |-5|=5

Problems with Absolute Value:

1. y=|x|

      Because x is in the absolute value sign it means that it can be either positive or negative, even if y is positive. That is so because the absolute value of x will always came out positive! But looking at the problem we can conclude that y cannot be negative because the absolute value of x will negative, even if it's negative in the brackets. 

ANSWER: x can be any number negative or positive, and 0, while y can only be 0 and all positive numbers.  And this equation is a function because there is only one input for each output, even though x=-2 and y=2 and then x=2 and y=2

2. A more complex one that is problem number 8. on page 151:

{ (x,y): |x|+|y| <= 1 }

      So like in the first problem x and y can be both negative and positive numbers, and they have to be less than or equal to 1. 

An easy number to always try first is 0 so if we plug in 0 for x and y they equation would look like: |0|+|0|<=1 and that statement is TRUE

Then we can plug in 1 for both x and y: |1|+|1|<=1 and that is TRUE also!

ANSWER: domain would be {x|-1,0,1}

Graphing with Absolute Value and Inequalities:

Problem #27 on page 151

27. y>=|x|

There are 3 things we need to figure out/do here:

1. What are the points that work? We need to find the boundaries.

2. Graph the inequality

3. Decide if this is a function.

Step 1:

When trying to find the boundaries it is always easiest to set the two side of the inequalities equal to each other. So the new equations would be:

y=x or y=-x   because x was in the absolute value brackets it can be negative or positive.

Step 2:

Now we need to see which points on the graph/which section solves the equation. We do this by picking a point from each section.

So if we plug in (3, 2): 2>=3 and that statement is FALSE!

Now if we plug in (0,-3): -3>=0 and that statement is also FALSE! 

If we plug in (-4,-2): -2>=|-4| = -2>=4 and that statement is again FALSE!

Finally if we plug in (1,4): 4>=1 and that statement is TRUE!!

The Final Graph:

The shaded are are all the values that work!!

Step 3:

This is NOT a function because for some inputs there is more that one output.

This information can be found in the textbook on pages 148-151.

And last but not least the homework for the weekend was on page151 #16 and 29!!