Tuesday, June 2, 2009

Transformations and Functions

Function: A set of ordered pairs in which each input (x) has at most one output (y).
Domain: set of usuable x's
Range: set of usuable y's


Vertical line test: in graph form, if any vertical line intersects the graph twice, it is not a function Odd Function: A function is odd if f (x) = - f (- x).
Even Function: A function is even if f (x) = f (- x).

Domain of a function
1. Polynomial Domain: [,] all reals


2.) restriction: stuff greater than or equal to 0


3.) restriction: stuff does not equal zero

Finding Domain and Range:
ex.
Domian:
set the y (f(x)) equal to 0 and then solve for x


0=

0=4-

= 4

x=2, x=-2

D: -2 x 2

D: [-2,2]

Range:
set the x equal to zero and solve for f(x)

f(x)=

f(x)= 2

R: 0 y 2

R: [0,2]

Graphs of a Function
*Increase as x value increases and y value increases

*Decrease as x value increases and y value decreases

*Constant as x value increases and y value remains the same

Relative Minimum and Maximum Values

-A function increases to its relative maximum value then decreases to its relative minimum then increases again. It also works the other way.

-On TI, maxima and minima can be found with 2nd Trace

ex.




Def: an odd function has the property f(-x)= -f(x). An odd function is symmetric to the origin. (x,y) is on the graph as well as (-x,-y)

Def: an even function has the property f(-x)= f(x). An even function is symmetric to the y-axis. (x,y) is on the graph as well as (-x,y)







Examples: Odd, Even or Neither
1.)

even





2.)



odd


3.)


f(-x)= x^2-x neither

Piecewise Functions: a domain-restricted function

ex. f(x)= x

f(x)= x ("v" shaped graph. Vertex at origin. You should know what this graph looks like)

f(x)= x -x less than 0 (picture graph with -x and x values)
x greater than or equal to 0


Transformations of function graphs
ex. f(x)=x+3

-f(x)= -(x+3)
-f(x)= -x-3 reflection over x-axis

f(-x)= -x+3 reflection over y-axis

3f(x)= 3(x+3) y-value times 3. x value stays the same
3f(x)= 3x+9


f(x-3)+1=... shift graph 3 units to the right and 1 unit up

f(x+1)-4 shift graph 1 unit to the left and 4 units down

Combination of Functions
Let f(x)= x g(x)=x^2

a.) (f+g)(x)
x+x^2
b.) (f x g)(x)
(x)(x^2)
x^3
etc.

Composition of functions

(f (sign looks like little o) g) NOT MULTIPLY!!!

f(x)= 5x-3 g(x)= 2x+5

ex. f(g(5))
g(5)= x-1
g(5)= 5-1
g(5)= 4
f(g(5))= f(4)
f(4)= 2(4)+1
f(4)=9

Decomposition of Functions

f(g(x))= (4x-12)^3

g(x)= 4x-12
f(x)= x^3


PRACTICE PROBLEMS


1. What is the domain and range of the relation y = 2x-3
2. Is the graph of (x-4)^2 - (y-1)^2 = 16 a function?
3. Determine the domain of f(x)= 2x/(x-3)
4. If f(x)= 4^x and g(x)= 2x-5 and f(g(x))=16, find x
5. Reflect the graph of y=x over the y-axis
6. What is the domain and range of y=x^2+2
7. Which of the following isnt a function? (a) y=x+1 (b) y=x^2+3x+6 (c) y=4 (d) x=0
8. Reflect y=3x-6 over the x-axis
9. If f(x)= 5x and g(x)= x^2 and f(g(x))=16, find x
10. What is the domain of f(x) = ?

ANSWERS:
1. All reals, all reals
2. no
3. all reals but x=3
4. x= 3.5
5. y=-x
6. all reals, y> (or equal to) 2
7. d
8. -3x-6
9. 4/5
10. x ≥ 4, x ≠ 8

7 comments:

Kate Nowak said...

I don't understand your "range" examples.

Stegs said...

we were still working on it

Kate Nowak said...

Hi Scott - Some graphs to illustrate the major transformations might be nice.

You need to have a few people read it and check for errors. For example, it's called a "piecewise" function, not "prewise".

You can use the "pipe" key to make vertical bars for absolute value. Like this: y = |x|. It's located on the right side of your keyboard.

Anna said...

This is really easy to follow. Good job! The only thing is that your missing the 10th practice question/answer.

kelseyyyy said...

This really gave me a better understanding about transformations and it reallyyy helped clear things up! Good job!

BFa said...

sorry anna, it has been fixed

Anna said...

hey guys i think the answer to number 9 is 4/(radical 5) not 4/5. can you recheck it?