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function notation difference quotient naming the domain of a function

FUNCTION NOTATION EXAMPLE: For the function G defined by G(x) = 2x2 - 5x, G(input) = 2(input)2 - 5(input) Evaluate: a) G(6) b) G(-1) c) G(a) d) G(b) e) G(h) f) G(x+h) Answers: a) 42 = 2( = 2( = 2( =2( =2( =2( b) 7 #2 c &B )2 - 5( )2 - 5( )2 - 5( )2 - 5( )2 - 5( )2 - 5( ) ) ) ) ) )

EXAMPLE: For the function defined by f(x) = 4x + 1, Write the corresponding ordered pair. f(input) = 4(input) + 1 a) f(3) = 4( ) + 1 = b) f(-1) =4( ) + 1 = c) f(a) = 4( ) + 1 = d) f(b) = 4( ) + 1 = e) f(h) = 4( ) + 1 = f) f(x+h) =4( Answers: )+1= +Ñ Ð $ ß " $ Ñ /Ñ Ð 2ß % 2 b " Ñ Ê( , ) Ê( , ) Ê( , ) Ê( , ) Ê( , ) Ê( , ) ,Ñ Ð c " ß c $ Ñ 0 ÑÐ B b - Ñ Ð +ß % + b 2ß % B b % 2 b " Ñ " Ñ .Ñ Ð ,ß % , b " Ñ

0 Ñ #B

b

% B 2 b

c &2

c) #+ c &+

.Ñ #, c &,

/Ñ #2 c &2

Exercises

Given the functions:

f(x) = kx - 2k, h(x) = 2x4 - 5x2 + 3,

g(x) = 3x - 5, k(x) = 4x

find each of the following: 1) f(5) 7) k(0) 12) h(a) answers: 2) f(-2) 8) k(3) 3) f(2) 9) k(-3) 4) g(1) 10) h(1) 5) g(0) 6) h(0)

11) g(a) (replace x with a) 14) f(x+2) 4) -2 11) 3a - 5 15) 4-x or ( 1 )x 4 15) k(-x) 5) -5 6) 3 7) 1

13) g(x-4) (replace x with x-4) 1) 3 8) 64 13) 3x - 17 2) 4 9)

1 64

3) 0 14) kxk 10) 0

12) 2a4 -5a2 + 3

REVIEW OF SLOPE Recall the slope formula for an line: In the linear function f(x) above This can be written using function notation:

slope = m = m=

13 - (-3) 3 - (-1)

y1 - y2 x1 - x2 16 4

=

=4

f(3) - f(-1) 3 - (-1)

For the two points (a,f(a)) (b,f(b))

f(a) the slope = m = f(b)--(a) b

For the two points (3,f(3)) (h,f(h))

f(3) the slope = m = f(h)--(3) h

For the two points (x, f(x)) (x+h, f(x+h)),

the slope = m =

f(x+h) - f(x) (x+h) - (x)

=

f(x+h) - f(x) h

This last expression for slope is called the DIFFERENCE QUOTIENT, h Á 0. It is used often in calculus to find the slope between two points of a function.

DIFFERENCE QUOTIENT

f(x+h) - f(x) h

h Á 0

Find the difference quotient for the function f(x) = 4x + 1: Solution: Break this difference quotient into pieces. f(x+h) = 4(x+h) + 1 = 4x + 4h + 1 Now subtract f(x) from this result.

f(x+h) - f(x) = (4x + 4h + 1) - (4x + 1) = 4h This is the resulting numerator. Put the pieces together.

oe %

Find the difference quotient for the function 0 Ð B Ñ oe c #B This is the resulting numerator. Put the pieces together.

oe c#

¨ ¨

oe

¨ 4 98¤ £51¨ 8¤ ¦ 4 ¦

¨ 4 ¦ 05§¤ 7&650©§¢ ¢ 4 ¦ ¨ ¦ ¤

oe

oe

¨ ¨ 0

oe

¨ # ¤ # ¦ ¨ ¦ ¤ &1(32$10§(

b $

¨ # ¤ ¢ # ¦ ¨ ¤¢ )$¦ ('&%$"!©¦ ¥

oe

oe

¨ ¤¢ ¡ ¨ ¦ ¤¢ ©§¥£¡ ¨ ¤¢ ¡ ¨ ¦ ¤¢ ©§¥£¡

Find the difference quotient for the function f(x) = 2x2 - x + 1

f(x+h) - f(x) h

è ë ë ë ë ë ë é ë ë ë ë ë ë ê [2(x+h)2 - (x+h) + 1] f(x+h)

h Á 0

f(x) è ë ë é ë ë ê (2x2 - x + 1) =

subtract all of the quantity f(x) = [2(x2 + 2xh + h2 ) - x - h + 1] - 2x2 + x - 1 =2x2 + 4xh + 2h2 - x - h + 1 - 2x2 + x - 1 = 4xh + 2h2 - h combine like terms

Now divide by the denominator h

f(x+h) - f(x) h

=

4xh + 2h2 - h h

= oe Put the pieces together. 0 Ð B Ñ oe % B b "

h(4x + 2h - 1) h

factor divide out the h

4x + 2h - 1

oe

% B

This represents the slope between any two points of the function f(x).

F F QP h Gpy F U E F C

oe

F b P C E h C U P b E F P C Gswv2tuXptxfP h £Gpv2U F U E F C y E h C

oe

F b P C E h C U P b E F P C P H h F E F C U Yxwvtugptsr1qapi2GE h RWU CA

b #2 c "

F H b E C 2g`fP h 2eYdXWIa`DYXIG$RWVT C UA P c b E H F E CA P hH F E CA U

oe

F H CA @ P H F E CA [email protected]

oe

EXERCISES For the functions f(x) = 2x - 3, g(x) = x2 + 4x - 3, find: 16) f(x+4) 20)

f(x+h) - f(x) h

17) f(x+4) - f(x) 21)

g(x+h) - g(x) h

18)

f(b) - f(a) b-a

f(x+4) - f(x) 4 g(b) - g(a) b-a

19) f(x+h) - f(x)

22)

23)

answers:

16) 2x + 5 21) 2x + h+ 4

17) 8 22) 2

18) 2

19) 2h 23) b + a + 4

20) 2

EXERCISES Evaluate the difference quotient for each function. Simplify. 24) f(x) = 1 - 3x 27) f(x) = 2x - 7 answers: 24) -3 25) 6x + 3h - 2 26) x(x+h)

-1

25) f(x) = 3x2 - 2x

26) f(x) =

1 x

27) 2

NAMING THE DOMAIN OF A FUNCTION Find the domain of each function: 28) f(x) = 3x2 - 2 31) f(x) = È x + 1 29) f(x) = 32) f(x) = È x2 - x - 2

x x2 + 1

30) f(x) =

x-2 33) f(x) = É x - 4

x x2 - 1

answers:

29) All real numbers

30) All real numbers except ,, 1 31) x 32) x 33) x Y -1 In interval notation: -1 Ò c" ß _ Ñ Ð c_ Ð c_ ß c" Ó r ß #Ó r Ò #ß _ Ñ Ñ

2 or x Y 2 or x > 4

34) -2 < x < 2

34) f(x) =

x 4 - x2

28) All real numbers

In interval notation: Ð c _ ß _ Ñ

ß _

Ñ

In interval notation: Ð c _

In interval notation: In interval notation: In interval notation:

Ð % ß _

Ð c #ß #Ñ

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