1

f x( ) = x4 − 3= x2 + 3( ) x2 − 3( )Zeroes :x = ± 34 = ±3

14

Domain : x ∈!{ } or −∞,∞( ) Range : f x( )∈! : f x( ) ≥ −3{ } or −3,∞⎡⎣ )

g x( ) = x + 6

Zero : x = −6Domain : x ∈! : x ≥ −6{ } or −6,∞⎡⎣ )Range : g x( )∈! : g x( ) ≥ 0{ } or 0,∞⎡⎣ )

1x −1

+ xx + 2

= 2

1 x + 2( )x −1( ) x + 2( ) +

x x −1( )x −1( ) x + 2( ) = 2

x + 2+ x2 − xx2 + x − 2( ) = 2

x2 + 2 = 2x2 + 2x − 40 = x2 + 2x − 6

Quadratic formula : x = −1± 282

= −1± 7

PreCalc Semester I Review KEY

Name: _______________________ Questions marked N are non-calculator problems 1. N Give the domain, range and zeros of: & .

2. N Solve for :

4( ) 3f x x= - ( ) 6g x x= +

x 1 21 2

xx x

+ =- +

2

log5 x( )+ log5 x + 3( ) = log5 4( )log5 x( ) x + 3( ) = log5 4( )5log5 x( ) x+3( ) = 5log5 4( )

x x + 3( ) = 4

x2 + 3x − 4 = 0

x + 4( ) x −1( ) = 0, x = −4,1

But log5 −4( ) does not exist, so x is only equal to 1

x = ±1,± 12,±3,± 3

2,±5,± 5

2

Root at x = 1

115 −26 13 −2

15 −11 2

15 −11 2 0

so f x( ) = x −1( ) 15x2 −11x + 2( )

Using the quadratic formula: x = 1130

± 121−12030

= 1230

,1030

= 25

,13

f x( ) = x −1( ) 5x − 2( ) 3x −1( ) so x = 1,25

,13

3. N Solve for : 4. Graph the function using an appropriate window.

a) Use your calculator to graph the function. What window is appropriate? b) List the possible rational zeros of the function.

c) Use your calculator to eliminate some of them. Then determine all the real zeros.

x 5 5 5log log ( 3) log 4x x+ + =

( ) 3 215 26 13 2f x x x x= - + -

3

f x( ) = x2 −5x −14x −1

=x − 7( ) x + 2( )x −1

f 0( ) = 02 −5 0( )−140−1

= −14−1

= 14, so the y-intercept is 14

Zeroes: x = −2,7

Vertical asymptote at x = 11

1 −5 −141 −4

1 −4 −18

so there is a slant (oblique) asymptote of y = x − 4

y 0( ) = 40e0.025 0( ) = 40, so the initial population is 40

y 50( ) = 40e0.025 50( ) = 139.614 = 140 or 139 is the population

at t = 50 to the nearest whole number

f t( ) = 10001+ 990e−0.7t = 530 ⇒ 530 1+ 990e−0.7t( ) = 1000 ⇒ 1+ 990e−0.7t = 1000

530

990e−0.7t = 1000530

−1⇒ e−0.7t =

1000530

−1

990

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟⇒ − 0.7t = ln

1000530

−1

990

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

t =

ln

1000530

−1

990

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−0.7= 10.025 days

5. N Given: , find

a) y-intercept

b) Zero(s)

c) Vertical asymptote(s)

d) Horizontal/slant asymptote(s)

e) Sketch the graph:

6. A certain population grows according to the model . Find the initial population and the population (to the nearest integer) when .

7. The spread of a flu virus through the population is modeled by the function ,

where is the number infected after days. In how many days will people be infected with the virus?

( )2 5 14

1x xf xx- -

=-

0.02540 ty e=50t =

( ) 0.7

10001 990 tf t

e-=

+( )f t t 530

4

8. N Sketch the graphs. Be sure to label all important points for one full period. a) y = 2 cos (x) a= 2

Amplitude= 2

b= 3

Period= 2𝜋

c= 0

Phase shift (start point)= No phase shift, starts at 1

d= 0

Vertical shift= None

b) 𝑦 = 𝑠𝑖𝑛 ()*

+,

a= 1

Amplitude= 1

b= -.

Period= )*/0

1= 6

c= *.

Phase shift (start point)= No phase shift, starts at 0

d= 0

Vertical shift= None

y = 2 cos (x)

𝑦 = 𝑠𝑖𝑛 32𝜋3 5

5

cosθ = 215, tanθ = 2

21= 2 2121

, cscθ = 52, secθ = 5

21= 5 2121

, cotθ = 212

f x( ) = 3sin 2x( )+ 2

5 feet

39°

x fe

et

cos 39°( ) = 5x

x = 5cos 39°( )

x = 6.433 feet

9. N If is a first quadrant angle, and , find the values of the other trigonometric functions of

. a)

b) c) d) e)

10. N What is an equation for this function?

11. A ladder is leaning against the side of a house. The base of the ladder is feet from the wall and makes an angle of with the ground. Find the length of the ladder. Be sure to draw a picture.

q2sin5

q = q

cosq = tanq = cscq = secq = cotq =

539°

2 5

√21

6

12ln x +1( )+ 2ln x −1( )⎡⎣ ⎤⎦ + ln x

12ln x +1( )+ ln x −1( )+ ln x

ln x +1( )12 + ln x −1( )+ ln x

ln x x −1( ) x +1( )12

⎛⎝⎜

⎞⎠⎟

ln x x −1( ) x +1( )

32° 20 miles

33°

tan 33°( ) = Ground to top of tower20

⇒ Ground to top of tower = 20 tan 33°( )tan 33°( ) = Ground to top of mountain

20⇒ Ground to top of mountain = 20 tan 32°( )

Height of tower = Ground to top of tower – Ground to top of mountainHeight of tower = 20 tan 33°( )− 20 tan 32°( ) = 0.491 miles = 2593.48 feet

12. N Write as the logarithm of a single quantity:

13. A tower atop a mountain is viewed from the plain 20 miles away. The angle to the top of the mountain is 32° and top the top of the tower is 33°. Find the height of the tower. Be sure to draw a picture.

( ) ( )1 ln 1 2ln 1 ln2

x x x+ + - +é ùë û

7

14. The population of a country (in millions) for the years 1983 through 1994 is given in the table where corresponds to the year 1983.

X 3 4 5 6 7 8 9 10 11 12 13 14 y 2.3 2.8 3.7 5.0 6.8 7.8 10.2 10.6 18.0 18.2 25.2 31.8

Use your calculator to graph a scatterplot. What window is appropriate?

Xmin= 0 Xmax= 15

Xscl= 1 Ymin= 0

Ymax= 32 Yscl= 2

a) Use the regression capabilities of your calculator to get a linear regression. Put it in y = 2.469x – 9.127, r2 = 0.883 b) Use the regression capabilities of your calculator to get an exponential regression. Put it in

y = 1.154(1.267)x, r2 = 0.991 c) Use the calculator to graph each of the models. Which is a better fit? The exponential equation: y = 1.154(1.267)x

d) Use that model to predict what the population will be in the year

y = 1.154(1.267)25 = 428.175 million

3x =

1Y ax b= +

2 xY ab=

2005

8

f x( ) = x4 + x3 + 3x2 + 9x −54, root at x = 3i

3i

1 1 3 9 −543i −9+ 3i( ) −9−18i( ) 54

1 1+ 3i( ) −6+ 3i( ) −18i 0

−3i

1 1+ 3i( ) −6+ 3i( ) −18i

−3i −3i 18i

1 1 − 6 0

f x( ) = x + 3i( ) x − 3i( ) x2 + x − 6( )f x( ) = x + 3i( ) x − 3i( ) x + 3( ) x − 2( )Zeroes: x = 3i,−3i,−3,2

P t( ) = 3000e0.12tP 4( ) = 3000e0.12 4( ) = $4,848.22

P t( ) = 3000 1+ 0.1212⎛⎝⎜

⎞⎠⎟

12t

P 4( ) = 3000 1+ 0.1212⎛⎝⎜

⎞⎠⎟

12 4( )= $4,836.68

15. N Find all the zeros of if one zero is . 16. If is invested at compounded continuously, find the balance after 4 years. 17. Use #16 again for compounded monthly.

( ) 4 3 23 9 54f x x x x x= + + + - 3i

$3000 12%

9

3x = 9

log 3x( ) = log9x log3= log9

x = log9log3

orx = log3 9

log2 x − log2 5= 3

log2x5

⎛⎝⎜

⎞⎠⎟= 3

x5= 23

x = 8( ) 5( ) = 40

f x( ) = x3 −5x2 + 6x

f x( ) = x x − 2( ) x − 3( )Zeroes: x = 0,2,3

Domain: x ∈!{ }Range: f x( )∈!{ }

5π4

+ 2π = 5π +8π4

= 13π4

2π9180π

⎛⎝⎜

⎞⎠⎟=2 180( )9

= 40°

Non-calculator problems 18) Solve. Show all steps a) b)

19) Given a) What are the zeros of the function?

b) Sketch the general shape of the function’s graph

c) What are the domain and range of the function?

20) Find the following:

a) A co-terminal angle of b) Convert into degrees

3 9x = 2 2log log 5 3x - =

( ) 3 26 5f x x x x= - +

54p 2

9p

0 2 3

10

A0ek 15( ) = 1

2A0

ek 15( ) = 12

15k = ln 12

⎛⎝⎜

⎞⎠⎟

k =ln 0.5( )15

= −0.0462

A 3( ) = 20e−0.0462 3( )

A 3( ) = 17.411 grams

8e2x + 3= 278e2x = 24e2x = 32x = ln 3( )x =ln 3( )2

= 0.549

6x+1 = 7x +1= log6 7( )x = log6 7( )−1x = 0.086

Calculator problems 21) A radioactive element has a half-life of 15 hours a) What is the decay constant?

b) If there were 20 grams of this element at the start, how much would be left after 3 hours?

22) Solve these equations. a) b)

28 3 27xe + = 16 7x+ =