function - texas a&m universitymayaj/142s19wir1completed.pdf · 2019. 1. 24. · 2x2 3x x5 1 3....

Math 142 Week-in-Review # 1 (Pre-calculus)

1. Determine if each of the following is a graph of a function. If the graph is that of a function, determine if that function has

an inverse.

(a)

(b)

(c)(d)

Passes the V - L test,

so

it is a function !

Fails the H - L test,

so does

not have aninverse !

Math 142 WIR, c�Maya Johnson, Spring 2019

2. Classify the following functions as a polynomial function, a rational function, a power/root, or as none of these:

(a) f (x) = 4x5 �2x2 +2x (b) f (x) = 3p

x (c) f (x) =2x ln(x2 �2x)

x4 �4

(d) f (x) = x1/3 +2x�4 +6x4(e) f (x) = ex2

(f) f (x) =2x2 �3xx5 �1

3. Write the following using interval notation: x <�3 or x � 3, but x 6=�4, x 6=�2 and x 6= 4

4. Starting with the parent function, f (x) =p

x, write the function, g(x), that results from reflecting f (x) about the x-axis,

contracting vertically by a factor of 6, shifting right 2 units, and shifting down 7 units.

2


5. Findf (x+h)� f (x)

hfor the following functions:

(a) f (x) = 4x2 �3x

(b) f (x) =1

x+2

(c) f (x) =p

x+2

3

aft x th )= 41×+42-3 l x th ) 4lxt④xh) - 3 x - 3h

= 41×2 thx thx the ) - 3 x - 3h

f- I x th ) - f L x ) = 4×2/+4hx t 4h x+442-34- 3h-4×2/+3×1fLxth)- = 8Kxt4K-3# =8×t4hh


6. Given the graph of f (x) shown below, find

a) f (�1), f (1), and f (2).

a) the value(s) of x such that f (x) = 1.

a) the zeros of f (x).

a) the y-intercept of f (x).

a) the domain of f (x).

a) the range of f (x).

4


7. Given the quadratic function f (x) = 2x2 �16x+24

(a) Find the vertex. Is the y-coordinate of the vertex a maximum or minimum?

(b) Find the zeros of the quadratic function.

(c) Find the domain and range of f (x).

8. Find and fully simplify the following.

2x+1

x+5�6

✓2� xx+6

◆

5


9. The owner of Tuff Toasters determines that if he sells a particular toaster for $37.50, then he can sell 32 units of this toaster

each day. If he decreases the selling price by $2.50, then he can sell twice as many toasters.

(a) Find the price-demand function, p(x) (assuming that it is linear), where x is the number of toasters sold, and p is the

price per toaster.

(b) Find the revenue function.

(c) Find the number of toasters sold that will give the maximum revenue. What is the maximum revenue?

(d) If the company has a fixed cost of $1,000 and a variable cost of $15 per toaster, find the company’s linear cost function.

(e) What is the company’s maximum profit?

6

• Az - 5/64, b -

- 25,

C = - 1000

Vertex :L h, k )

k=P( 160 ) I tooo

h= - b- =

-25=160MaxProfitis$4oo

Za 4-5/64)


10. Use the Laws of Exponents to fully simplify the following, and express the answer without using radicals or negative

exponents.

a)5

r⇣15a�6b5

5a3b�5

⌘�3

b)

⇣15a�5b4

5a3b�4

⌘�3

b)30x�2

py4z8

6y�5z 3p

x

7

( T.fi?b4-t4D..6a-s.b53=3

'am

. b-"

= ,aj =g"

30 x- 2

( y4 zs )

' k

Tjzxs= =

5×-2-43 yz- t - 5) z4

- I

= 5×-7/3y7z3

=5¥


11. Use properties of logarithms to fully expand the following into the sum or difference of simpler logarithms.

(a) log3

✓z+4

32

◆

(b) ln(xpy)

12. Express each of the following as a single logarithm.

(a) lnu�2ln6

(b) ln4+6lnx� 1

3lnz6

13. For the given functions f (x) and g(x) below, find ( f �g)(x) and (g� f )(2).

(a) f (x) = log2(x) and g(x) = 2

x2+x

8

Indy" 2) =

In x they" 2

=hxttz

t.nu - 2 hub = Inu - tub =hru-ln①

flglx) ) .. log ! 547) -

- tog" " )

=×2

Gtf 127) = zHHtH?21105%25+165%4

= z' t '

= 22


(b) f (x) =p

x and g(x) = (x+1)2

14. Solve the following equations exactly for x, if possible.

(a) log2(x)+ log

2(x+2) = 3

(b) 3e2x+5 = 6

(c) 3e2xx2 �12e2x = 0

9


(d) 4x�1

23x = 8

4

(e)1

25= 5

4x · 1

53x2

(f) x(x�1) = 6

(g) 3x3 +6x2 �9x = 0

10

÷ =54 ×

.

5-3×23×2 - 4×-2=0

Cannot factor.

Use Quadratic Formula: × = -bIFza

I'

-

-E" " " "

×=4±f=4tf=4±Fy = 4±2y=2t- 2 = 4×-3×2 3

⇒

x=2audx=2?x'

- X = 6

XZ - X - 6=0

( x - 3) ( x t 27=0

⇒

x=3andX=

3 x ( x'

t 2x - 3) = O

3 x ( x t 3)( x - I ) = O

- - -

⇒ X=O,×=-3,andX=t€


15. Use the properties of logarithms to find and fully simplify the following.

(a) log3(92)

(b) log6( 5p

6)

(c) 99log

9(41/9)

16. The population of a particular city grows continuously at a relative growth rate of 5.4%. If 30,000 people currently live in

the city, what will be the population in eight years?

17. If you invest $3500 into an account now that earns interest at an annual rate of 4.6% compounded monthly, how much

money will be in the account after 10 years? (Round your answer to the nearest cent.)

18. Given logb 3 = 0.6826 and logb 4 = 0.8614, find logb�

16

3b3

�

11

lost 325=105434

10*45 =

BBBEBEEBEGGBBFqlysgl4

" 9)Fth 449 )9= ④

A =P ( I t IT ! P a z soo ,r =

.046

, m = 12 (monthly), t = to

.

A = 3500 ( I t

.0,4¥)" " "

= 5539 . 39. $5i53

↳ Sb (E) = log 66 - log b3B -

- log 512 - tog - 105%5

= 2 log b4 - log § - 3--21.8614) - I . 6826 ) - 3 = -1.9598J


19. Find the zeros of f (x) =x2 �6x+8

x+3

20. How long will it take a $300 investment to be worth $1500 if it is continuously compounded at 9% per year? (Round to the

nearest year.)

21. Evaluate f (2), f (�1), f (�3) and f (4) for the following piecewise defined function.

f (x) =

8>><

>>:

2+ |3� x|x+3

if �5 x �1

x2if x = 2

log4(x)�2 if x � 3

22. An electricity company charges its customers a base rate of $11 a month, plus 8 cents per kilowatt-hour(kWh) for the first

1300 kWh and 10 cents per kWh for all usage over 1300 kWh. Express the monthly cost E as a function of the amount x of

electricity used.

12

XZ- 6×+8

- = o ⇒ XZ-6×+8=0×+3⇒ ( x - 2)( x-47=0

⇒ x=2adX=

A- . Pert,

p .

. 300,

A -- 1500,5=09

. Ins =.

09T

09T 7097091500=300 e

. 09T

Too To In}:# at t=l8y

fth -125=40

ft - I)=2tf?I=2Hz4I=I=③H-H-DNECY.IE?nisfc4y=iog#D

-2--1-2

First 1300kWh ( OEXEBOO ) :

:÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷:


23. Rewrite the following functions as piecewise-defined functions.

(a) f (x) = x+ |3x�4|

(b) h(x) =3+ |2�3x|p

2x+1

24. Sketch a graph of the following piecewise defined function.

f (x) =

8><

>:

�3x�10 if x <�2

�2x+2 if x =�2

x�3 if x >�2

13

.Solve 2 - 3×70 for x :

2 - 3 x z O

-

II'-

-

I hlx ) -

.

{'

iif x 543

X I 2- I t3×34,7¥,

if x 743

For x L - 2 :

closed circle

-31-2 ) - 10=-4 opencircle

•

For x =- 2 :! Y -

- - 21-2 ) +2=6 closed point÷For X > -2 '

-

open circle

O - 3 =-3closed circle


25. Find the domain of the following functions:

(a) f (x) =ex2+2x

ln(x�1)p

8� x

(b) f (x) =6p

x�4

(x+2)(x�8)

(c) f (x) =log

7(x�4)

(x+2)(x�8)

14

Top : 874 even root,

x - 470 ⇒ x 24

Bottom : txt 2) ( x - 8) to ⇒ × # - 2 and x # 8. Ignore x= -2 since X 74 -

Combine ; LEXIDomainil4.HU/8#y

TOP '-

105,1×-4 ).

⇒ × - 470 ⇒ X > 4

Bottom :( × + 2) ( × - 8) to ⇒ Xt - 2 -d K¥8

. Ignore x --

-2again since × , 4

.

Combine ; L#µ Domaon:l4,8)Ul8,o


(d) f (x) =

8<

:

2

x2 �9x+14if 0 x 4

4x+5 if x > 4

(e) f (x) =

8>>>>>>>>>><

>>>>>>>>>>:

px�1 if 0 x < 4

x ln(x)5

p(x�5)(x�6)

if 4 < x 7

x2 +2

ln(12� x)if x > 7

15

0

gNo equality means x -

-4 is not in the

domain,

or Xt 4.

-

Foo o E x a 4 : TE , ⇒ x - I 20 ⇒XFor 4 L x 17 :

Top : In ( × ) ⇒ XBottom : ( x - 5) Cx - 6) to ⇒¥5and

For x > 7 :

Top : × 2+2 L - oo ,• )

Bottom : lnllz - × ) # O ⇒ 12 - x t I ⇒

x¥::÷÷÷÷¥÷÷i÷÷:::÷÷...u

function - texas a&m universitymayaj/142s19wir1completed.pdf · 2019. 1. 24. · 2x2 3x x5 1 3....

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