1. Functions
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2. -To identify a function. -To determine domain & range
Objective:
-To identify a function.
-To determine domain & range
-To graph functions
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3. What is a relation ? A relation in math is a group of ordered pairs.
{ ( 4, 5), (6,7), (9,10)}
{ (1,1), (1,2), (1,2) }
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4. A function is a relation in which no two ordered pairs
have the same x-value.
{(3,2), (4,5), (6,-1), (0,2)} is a function because all
the x-values are different.
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5. D R x y A function is the set of (x,y) such
book definition D R x y
A function is the set of (x,y) such
that X is an element of the domain
D and Y is the corresponding
element of the range R.
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6. This is formally written { (x,y) l y = f(x) }D R
x maps
onto y y x
This is formally written
{ (x,y) l y = f(x) }
the set
such that
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7. A function -no two ordered pairs have the same x-value.
What’s NOT a Function?
{(-3,2) ,(4,5) , (6,-1) ,(4,2)} is NOT a function.
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8. Range - set of y values. {(3,2) , (4,5) , (6,-1) , (0,2)}
Domain - set of x values.
Range - set of y values.
{(3,2) , (4,5) , (6,-1) , (0,2)}
Domain {0,3,4,6}
Range {-1,2,5}
Arrange order.
Do not repeat numbers.
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9. Range - set of y values. {(3,2) , (4,5) , (6,-1) , (0,2)}
Domain - set of x values.
Range - set of y values.
{(3,2) , (4,5) , (6,-1) , (0,2)}
Domain {0,3,4,6}
Range {-1,2,5}
{ } means
the set of
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10. A vertical line only intersects 1 point.
What is a Function?
A vertical line only intersects 1 point. Y X
passes the
vertical line test
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11. What’s NOT a Function? If the vertical line intersects 2 points,
the graph is NOT a function. Y X
Fails the vertical line test
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12. The value of y depends on x
What’s a Function
The value of y depends on x x
function
Output y
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13. Y is the Dependent variable.
Since Y DEPENDS on X,
Y is the Dependent variable.
X is the
Independent variable
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14. same as f(x) = 3x +2 read: the function of X Let’s let y = 3x + 2
The number you substitute for X
determines the number you get for Y.
same as f(x) = 3x +2
read: the function of X
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15. Can also be written as g(u) = 3u + 2 Read g(u)= The function g of u =
Function Notation
y = 3u + 2
Can also be written as
g(u) = 3u + 2
Read g(u)=
The function g of u =
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16. Function Notation (x,y) is a solution of y = 3x + 2.
(x, f(x)) is a solution of f(x) = 3x + 2.
f(x) is the same thing as y.
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17. The bike’s
height is a
function of
what ?
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18. Function Notation f(1) = 3(1) + 2 f(1) = 3 + 2 f(1) = 5
If x = 1 and f(x) = 3x + 2, then:
f(1) = 3(1) + 2
f(1) = 3 + 2
f(1) = 5
f (1) means use 1 for x
Remember
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19. h(x)= - 2x2 -x Find h(-3) h(-3)= - 2(-3)2 - ( -3) h(-3)= - 2(9) +3
Try this one
h(x)= - 2x2 -x
Find h(-3)
h(-3)= - 2(-3)2 - ( -3)
h(-3)= - 2(9) +3
h(-3)= -15
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20. Function Notation f(x) = 3x2 + 2x f(x+h) = 3(x+h)2 + 2(x+h)
f(x+h) = 3(x2+2xh +h2) + 2(x+h)
= 3x2+ 6xh +3h2 +2x +2h
Be careful
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21. f(x) = 3 X g(x) = x+1 find f ( g (x)) Main function is f(x) f(x)= 3 X
= 3 ( ) What replaces X?
(x+1)
f(g(x)= 3x + 3
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22. f(x) = X 2 g(x) = x-2 find f ( g (x)) f(x)= X 2
Main function is f(x)
f(x)= X 2
= ( ) 2 What will replace X?
(x-2) 2
= (x-2)(x-2)
= x2 -4x + 4
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23. Finding the domain and range
What x values can be used in the function f(x) = 2x + 3 ?
Any real number.
So D = set of reals.
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24. What is the result for the y values? They are also real numbers.
f(x)= 2x+3
What is the result for the y values?
They are also real numbers.
So R = set of reals.
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25. What x values can be used in the function f(x) = 3 / x ?
Any real number except 0 can be used. So D = set of reals, x = 0.
What set will result for the y values?
These are also real numbers except 0. So R = set of reals, y = 0.
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26. What x values can be used in the function f(x) = √ x ?
Only + reals and 0 can be used. Why? So D = {x > 0 } .
What set will result for the y values? These are also 0 or larger So R = { y > 0 } .
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27. Find the domain and range
x - 3 = 0
D = {reals x = 3}
R = {reals y = 0}
x + 2 > 0
D = { x > - 2 }
R = { y < 0 }
f(x) = _4_
x - 3
f(x) = -√ x + 2
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28. Find domain and range y = x2 - 2 D = all real nos. R > -2
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29. Find Domain and Range
Domain
-4 < x < 3
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30. Find Domain and Range
Range
-1 < y < 2
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31. D = { x l -3< x < 4 } Piecewise Function Find domain
x values are
from -3 & 4
-3< X < 4
D = { x l -3< x < 4 }
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32. Piecewise Function Find range Y values are -1, and 1 to 4
y = -1 or 1< y< 4
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33. x - 4 < x < -1 f(x)= x2 x > 0 2 different equations
Graph Domain
x < x < -1
x x > 0
f(x)=
2 different
equations
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34. x - 4 < x < -1 x2 x > 0 f(x)= Graph one at a time. f(x) = x
domain
x < x < -1
x x > 0
f(x)=
Graph one at
a time.
f(x) = x
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35. x - 4 < x < -1 x2 x > 0 f(x)= f(x) = x2 Graph domain
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36. Determination can conquer almost anything.
Thought for the day
Determination can conquer
almost
anything.
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