Math 121 Exam I

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Math 121 Practice Exam #3, spring ’10 WITH SOLUTIONS

You MUST show work for all problems to receive full credit.

1.

For the given functions

f

(

x

)  2

x

5  1 and

g

(

x

)  3

x

1  1 find Name ____________________

a)

(

f

g

)(

x

) . Then find the domain of (

f

g

)(

x

) . (

f

g

)(

x

)  15

x

1   3

x

5 ;

D

f

g

:   

x x

  1 3 ,

x

 1 3

b)

(

g

f

)(

x

) . Then find the domain of (

g

f

)(

x

) . (

g

f

)(

x

)  2 2

x x

  1 14 ;

D g

f

:

x x

 1 2 ,

x

  7

2.

For the given function

f

(

x

)  3

x

5

x

 1  2 find

a).

the inverse function

f

 1 (

x

) .

f

 1 (

x

)  2

x

5

x

 1  3

b).

the domain and range of

f

(

x

) and

f

 1 (

x

) .

D f

:

x x

 2 5 ,

R f

:

y y

 3 5 ;

D f

 1 :

c).

X-int., Y-int. of

f

(

x

) and

f

 1 (

x

) .

x x

 3 5 ,

R f

 1 :

y y

 2 5

f

(

x

) : X-int: 1 3 , 0 , Y-int:

d).

V.A., H.A. of

f

(

x

) and

f

 1 (

x

) . 1 2 ;

f

 1 (

x

) : X-int: 1 2 , 0 , Y-int: 1 3

f

(

x

) : V.A.:

x

 2 5 , H.A.:

y

 3 5 ;

f

 1 (

x

) : V.A.:

x

 3 5 , H.A.:

y

 2 5 ; ;

4.

[10 pts] Graph the following function, by first showing the basic function, then applying the appropriate transformations to it, using the graphs below to show each stage. Then, state the domain, range, asymptote, and compute algebraically all intercepts for

f

(

x

)   3

x

 1  1 The basic function is

f

(

x

)  3

x

Transformations: a) shift right 1 unit; b) reflection about X-axis; c) shift up 1 unit.

y

Domain of

f

(

x

):___all real numbers______ Range of

f

(

x

):_____y<1_______________ Y-int:__(0,2/3)____ X-int:__(1,0)_______ Asymptote:____(H.A.)__y=1______________

5.

[10 pts] Graph the following function, by first showing the basic function, then applying the appropriate transformations to it, using the graphs below to show each stage. Then, state the domain, range, asymptote, and compute algebraically all intercepts for

f

(

x

)  log 2 ( 

x

 2 )  1 . The basic function is

y

 log 2

x

Transformations: a) shift left 2 units; b) reflection about Y-axis;

x

c) shift up 1 unit;

y

Domain of

f

(

x

):__x<2_________________ Final graph Range of

f

(

x

):__all real numbers________ Y-int:__(0,2)___ X-int:__(3/2,0)____ Asymptote:__(V.A.) x=2_____________

x

6.

[10 pts] Change each exponential expression to an equivalent logarithmic expression: 1 .

3

x

 9 ;  log 1 .

3 9 

x

; 1 3  3  1 ;  log 3 1 3   1 ;

e m

 7 ;  ln 7 

m

;

7.

[10 pts] Change each logarithmic expression to an equivalent exponential expression: log

a

3  4 ; 

a

4  3 ; ln 5 

y

; 

e y

 5 ; log 2

x

 5 ;  2 5 

x

;

8.

Write a following logarithmic expression as a sum and difference of logarithm. Express all powers as factors. log 3   3 (

x x

2 2 ( 

x

2  )(

x

1 ) 3  1 )   ,

x

 1 .  1 3 log 3 (

x

2  2 )  1 3 log 3 (

x

 1 )  2 log 3

x

 3 log 3 (

x

 1 )

9.

Write this expression as a single logarithm. 2 (

x

 1 )  (

x

 2 ) 2 1 2 ln(

x

 1 )  3 ln(

x

 2 )  2 3 ln 8  1 3 ln(

x

 2 ) ,

x

 2  ln 4  3

x

 2

10.

Solve each equation:

a).

2

x

8 

x

c).

9 2

x

d).

e x

 27 3

x

 4 ; 

x

 12 5 ;  5

e

x

 2 ;  3

x

 0 ; and

x

 ln 5 ;

e).

3  2 3

x

 1

f).

e x

 1  4

x

; 

x

 0 ;

b).

(

e

3 )

x

 1

e

4  

x

2 ; 

x

 2  2 ln  ;

g).

e

x

2 3 2

x

; 

x

  2  3

x

 1 5 ; 

x

   4

;

x

 1

;

log 2 27 ; 

x

 3   3  1 1 ;

k).

(

e

x

) 2

x

 1

e

5 

e

4

x

e

x

e

; 

x

 1 2 ; and

x

11.

Solve each equation:

a).

f

(

x

)  log 3 (

x

 1 )  4 ; 

f

( 8 )   2

;

  3 ;

b).

log 5 (

x

2  9 )  log 5 (

x

 3 )  2 ; 

x

 28

; c).

log

x

4   2 ; 

x

 1 2 ;

d).

log 2 ( 3

x

 2 )   2 ; 

x

 17 8

; e).

1 2  log 4 (

x

 4 )  log 4 (

x

2  1 )  1  log 4 (

x

 1 ) ; 

x

 9

;

x

 0

is extraneous.

2 4

12.

A child’s grandparents purchase a $10,000 bond fund that matures in 18 years to be used for her college education. The bond fund pays 4% interest compounded semiannually.

a).

How much will the bond fund be worth at maturity? 

A

 10 , 000  1 .

04 2  18  $ 20 , 398 .

87

;

2

b).

What is the effective rate of interest? 

r eff

a) –9 b) 7 c) 8 d) 4

x

2  4

x

 1  4 .

04 %

; c).

How long would it take the bond to double in value under these terms? 

t

 ln 2 .

04  17 .

5

years

;

2 ln  1 2

13.

The half-life of radioactive carbon14 is 5600 years. A fossilized leaf contains 70% of the carbon14 that it originally had. How old is the fossil?

t

 ln(.

7 )

r

 ln(.

7 )  5600 ln(.

5 )  2881 .

6

years

; 14.

A culture of

Salmonella

bacteria is started with 0.01 gram and triples in weight every 16 hours.

a).

If the weight of the culture has been growing exponentially, how many grams will be after 25 hours?  first we have to find the rate : 3 

e r

 16

;

r

 ln 3

;

16

A

 .

01 

e

25  ln 3 16  .

05566 grams;

b).

How long will it take for the culture to weigh 0.27 gram? .

27  .

01

e

ln 3 

t

16

;

t

 16  ln( 27 ) ln 3  48 hours.

15

. Make one choice below to complete the sentence. The graph of the basic exponential function a) has a vertical asymptote b) increases when

x

< 0 and decreases when

x

> 0. c) has a maximum value of 1 d) has an unrestricted domain

d 16

..If

f

(

x

) 

x

2  1 and

g

(

x

)  2

x

 1 , then

f

(

g

(  2 )) 

c

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