Precalculus Term 1 Review

Transformations

100

Write the equation that translates the absolute value function three units to the right.

|x-3|

200

What transformations occurred on this parent?

y = -3(2x-9)²-8

Right 9

Horizontal shrink by 1/2

Vertical stretch by 3

Reflect Over x-axis

Down 8

300

Write the equation that shows the exponential function reflected over both axes and shifted up four.

y = -e^-x+4

400

A horizontal shrink by 1/2 on the squaring function is equivalent to what other transformation on the squaring function?

Vertical stretch by 4.

500

Sketch the function: 

3(0.5x-1)-2

Graph in calculator to confirm.

Compositions

100

Given: f(x)=3x+2 and g(x)=5x

What is the simplified composition g(f(x))?

15x+10

200

Given f(x)=3x²  and g(x)=sqrt(3x+9)

Find f(g(x)) and state the domain.

f(g(x)) = 9x+27

D: [-3, inf)

300

Given: f(x) = log(x) and g(x)=5x-12

Find: g(f(1000))

3

400

Given: g(x)=ln(x) and f(x)=e^2x+10

^{Find g(f(x)) and state the domain.}

g(f(x)) = 2x+10

D: (-inf,inf) (think about this - it's not because 2x+10 has domain of all reals.)

500

Given: f(x)=log(x), g(x)=5x³+680, and h(x)=|6-5x|

Find: h(f(g(4))

9

Logs & Exponentials

100

What is the mistake in the following: 

10 = 3e^2x+1

10/3 = e^2x+1
​

log(10/3) = (2x+1)log(e)

log(10/3) = 2x+1

x = ( log(10/3) - 1 ) / 2

log(e) does NOT equal one - so in line 4, they should have divided by log(e) and their final answer should have a log(e).

Note: Using log vs. ln is NOT the mistake - you can always use either. The issue is in the simplification.

200

Write a function for the situation: 

A population of 18000 bees is decreasing by 3.8% every six years. Where p(t) is the population at t years.

p(t) = 18000(1-.038)^t/6 or equivalently 

p(t) = 18000(0.962)^t/6

300

What is log₂₅(5)?

1/2

400

How long does it take for 80% of a radioactive substance to decay if it's half life is 425 years?

Round your answer to the nearest thousandth.

986.819 years.

500

In Burlington, WI - there used to be a place called the Sci-Fi Cafe. It was weird. Ms. Dexter went once with some high school friends and was told that you could summon fairies (well, only women could - sorry, guys) by putting out rings of M&Ms (this is not a joke - this really happened to her.) Ms. Dexter lays out a huge ring of M&Ms. Initially, she finds 10 fairies (this part did NOT actually happen). 4.5 weeks later, there are 83 fairies. Assuming exponential growth - what would be the r value in the model P=Poe^rt model for this situation? Give 6 decimal places!

r = 0.470279

Solving Equations

100

Solve: 

81^2x+1 * 243^x =3⁵

1/13

200

Solve for x. Give your solution as a decimal rounded to the nearest thousandth.

4 * 15^6x+9 - 5 = -3

-1.543

300

Solve: -9log(-x-10)=-36

-10010

400

Solve: log(2x²+8) + log(4) = 3

11 AND -11

500

A population (in millions) is decreasing such that:

p(t) = 500 / (1 + e^-0.065t) where t is measured in decades.

In how many years will the population reach 300 million? Round to the nearest hundredth.

62.38 years.

Miscellaneous

100

If a function H gives a person's height off the ground in feet when skydiving at any time t. What would H inverse do?

H inverse would be able to take a person's height off the ground and give the time that they've been falling. 

200

What is the domain for: 

3ln(x+2) / (x-2)

(-2,2) U (2,inf)

300

What is the domain & range for: 

y = 3ln(2x-8)+10

D: (4, inf)

R: (-inf, inf)

400

Given f(x) = log(3x-9) + 10

Find f^-1(y) AND give domains for both f and f^-1

f^-1(y) = 1/3(10^y-10+9)

D of f: (3, inf)

D of f^-1 : (-inf, inf)

500

The Loggers stadium has 6000 seats and sells tickets for $13. At this price, they sell about 4,800 seats. Market research says that for each 50 cent decrease in price, they'll sell 50 more seats. Write an equation for the profit AND state the relevant domain.

Equation: (4800+50x)(13-0.5x)

D: (-96, 24) *Seat capacity