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Revison Chapter One

1. Thefunction fg are definedby :

f(x )=3x , x>0 g (x )=x1,

x>1

aSketch the graphs of fg .

bFind f1g

1, statingthe domainsranges of f

1g1

c Determine whether f ggf exits.

Ans b f1 (x )=ex

3 , Df1={x :xR }R

f1={y :y>1 , yR }

g1 (x )=x2+1 , D

f1= {x :x>0 , xR }R

f1={y :y>1 , yR }

c) f g exitsg f doesnot exit

2. The function fg are defined by :

f(x )= (x+1) , x>1 g (x )=x2+2 , , xR , x>0

aShow that f has aninversefunction f1

,sketch onthe same axes ,

the function ff1

.

bDetermine whether f gg f exits. State your reason.

Ans : a) f1 (x )=ex1

b) f g exitsg f does notexit

3.. Thefunction fg are definedby :

f(x )=3x5 , xR g (x )=e3x , xR

a State the range of g .

bFind f1g1 ,Then , sketchf1g1 onthe same axes.

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c State , giving a reason, the number of roots of the euation f1 (x )=g1 (x ) .

d !va"uate fg (1

3 , giving your answer correct3decima"s p"aces.

Ans : aR

g={y :y>0 , yR

b

f1 (x )=1

3(x+5 ) g1 (x )=

13

x

c has on"y one root . intersect at one point .

d 0.843

4 . . The function fg are defined by

f(x )=1

x, xR x #0 g (x )=2x1 , xR

Find f gitsdomain

Ans : f g (x )=

1

2x1x #

1

2

Domainof f g {x :xR , x #12}

5. . The function fg are defined by

f(x )=1x2 ,1$ x $1 , g (x )=x2+2 , xR

Find a g fb state whethet f g is exits .%hy &

Ans :3x2, 'ot exits

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6. The function f , gh are defined by :

f(x )= x

x+1, g (x )=

x+2x

, h (x )=3+2

x,

aState the domain of fg .

bFind g fstateits its domainrange

c State the domainrange of h

d State whether h=g f .(ive a reason for your answer .

a) {x :xR , x #1} g={x :xR , x #1 }

b)3+

2

x, x #0

D = {x :xR , x #0 , x #1 }R={y :yR , y #1,y #3 }

c) {x :xR , x #0 , } {y :yR , y #3 }

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