MAC1140 FINAL EXAM QUESTIONS OP

You are reviewing a model your company has to calculate the cost to modify one of your software packages for individual usage. You need to update the model due to inflation and changes in the industry, and you have developed a transform that does exactly this. In particular, the updated cost is calculated by 𝑈(𝑥) = 1.2𝐶 (𝑥) − 250 where 𝐶(𝑥) is the original cost. If a customer had originally been quoted a cost of $500 for modifying 3 software packages, what would the updated cost be?

$350

Which of the following denotes a point that is left 2 and up 2 from the origin?

(-2,2)

Which coordinates describe a point that is 5 units up and 2 units to the left of the origin?

(-2,5)

What is the domain of the real-valued (ie no non-real numbers allowed) function f(x) = 2 √-x + 1 + 7 ?

(-∞, 1]

Consider the following rational function: 𝑟(𝑥)= x⁴-x³-13x² +25x -12 / (x - 1)² Which of the following are coordinates of holes on the graph of r(x)? (Select all that apply)

(1,-10)

Consider the following rational function: 𝑟(𝑥)= 𝑥²+3𝑥−10 / 𝑥²−7𝑥+10 Which of the following are coordinates of holes on the graph ofr(x)? (Select all that apply)

(2, ⁻⁷⁄₃)

Which of the following is equivalent to the expression √125x³y⁶ - √5x ?

(5 | 𝑥𝑦 ³| - 1) √5x

Fully factor the following polynomial: 10x² −8x−18

(5x−9)(2x+2)

What is the fully factored form (with real coefficients) for the polynomial: f(x)= x⁴ - 256

(x-4)(x+4)(x² +16)

Fully factor the following polynomial: 9x⁶ - 64

(³√3x−2) (³√3𝑥+2) (³√9x²+2 ³√3𝑥+4) (³√9x² - 2³√3𝑥+4)

What is the domain of function: f(x) = 1n (x + 1) +12

(−1,∞)

Consider the following rational function: 𝑟(𝑥)= 𝑥²+2𝑥−15 / 𝑥²+9𝑥+20 Which of the following are coordinates of holes on the graph of r(x)? (Select all that apply)

(−5,8)

What is the domain of the real-valued function 𝑓(𝑥) = −13 ⁶√-3x+17-1

(−∞, ¹⁷⁄₃]

What is the domain of function: 𝑓(𝑥)=log₂(−3𝑥+6)−1

(−∞,2)

What is the domain of the radical function f(x)= ³√-x-1

(−∞,∞)

Consider the function 𝑝(𝑥)= −3𝑥¹² −4𝑥¹⁹ + 3𝑥⁹ +2𝑥. Which of the below values are a possible number of absolute extrema for this function? (Select all that apply!)

0

Consider the function 5x¹³−10x⁸+7x¹⁵+x¹⁹. Which of the below values are a possible number of relative extrema for this function? (Select all that apply!)

0, 12, 18

A particular isotope has a half life of 7 years (that is; the amount of material remaining is cut in half every seven years). How long would it take before three quarters of the starting material has decayed?

14 years

Consider the polynomial: 𝑝(𝑥)= 14𝑥³+ 7𝑥² + 9𝑥 +323. Which of the following can not be a root of the polynomial? (Hint: Use Rational Root Theorem)

17𝑥−7

Consider the polynomial, 𝑝(𝑥) = −13𝑥¹² − 4𝑥¹⁹ + 3𝑥⁹ +2𝑥 How many zeros does this polynomial have including complex valued zeros and counting multiplicity?

19

Which of the following is equivalent to log₅ (49). Hint: Use change of base formula.

2 / log₇ 5

When someone calls you and they are in your contacts, your phone uses their phone number to display their name. What are the domain and codomain of this relationship?

Domain: phone numbers in your contacts; Codomain: names

Which of the following equations are polynomials? I: 𝑓(𝑥)=𝑥²+1 II: 𝑔(𝑥)=13𝑥³ +7 III: ℎ(𝑥)=7 IV: 𝑘(𝑥)=𝑥³ +3𝑥^1/3 +1

I, II, and III Only.

What is the primary difference between a jump discontinuity and a hole discontinuity?

Jump discontinuities have a vertical space between the left and right segments of the graph whereas a hole is only missing a point, but the left and right sides otherwise line up.

A university parking lot designates spaces based on parking permit colors. The lot has 6 rows of 12 parking spaces each. The rows are labelled A-F and the spaces are numbered 1-12 in each row. The different color permits are allowed to park as follows: Red permits may park in any space. Orange permits may park in any space in rows B-E. Green permits are allowed to park in row E spots 6-12 and (any spot) in row F. Blue permits may only park in row D. Is the relationship that determines your potential parking spaces based on your permit color a function?

NO.

At the Friends of the Library book sale, the books are arranged into a number of categories based on their value as follows; Tier 1: $2.50 Tier 2: $5.00 Tier 3: $10.00 Consider the relationship that tells you all the possible books you can buy based on how much you want to pay. Is this relationship a function?

No, for one price you can buy many different books.

Consider the following rational function: 𝑟(𝑥)= 𝑥²+x-2 / 𝑥²- 3x - 10 Which of the following would be best modeled by logarithms? (Select all that apply)

None of these are correct

Consider the polynomial 𝑓(𝑥)= 𝑥² + 3𝑥 +2. Which of the following statements are true? I: 𝑓(𝑥)f(x) has 2 roots but no real zeros. II: 𝑓(𝑥)f(x) has 2 roots and 2 real zeros. III: 𝑓(𝑥)f(x) is degree 2 which is why it must have 2 real zeros. IV: 𝑓(𝑥)f(x) is degree 2 but has no real zeros

Only II is correct.

Consider the following rational function: 𝑟(𝑥)= 7x³ +18x² +20x + 8 / 2𝑥⁴ +𝑥²−3𝑥−10

The Horizontal Asymptote is y = 0

Consider the function 𝑓(𝑥)= [ −2𝑥−8, −2 ≤ 𝑥 ≤2 [ 3 √ | x-1 | -15, 2 < 𝑥 ≤6 [−4(𝑥−1) 2 + 2 , 6 < 𝑥 ≤ 12 Determine if f(x) is discontinuous at 2, and if so, what type of discontinuity.Hint: Graph the function to determine the answer!

The function is continuous at 2.

Consider the function 𝑓(𝑥)= [5(𝑥+1)³+5, −1≤𝑥≤3 [−4𝑥+3, 3<𝑥≤6 [𝑥−1, 6<𝑥≤7 Determine if f(x) is discontinuous at 4, and if so, what type of discontinuity. Hint: Graph the function to determine the answer!

The function is continuous at 4.

Which of the following has an absolute maximum?

The graph that is an upside down U and the top of the U is touching the graph.

What does it mean to increase logarithmically?

The more of something you have, the slower it increases.

Simplify the expression ⁴√x⁴ - 16

This is already as simplified as it can be

Simplify the expression √x⁴-81

This is already as simplified as it can be

Simplify the expression ³√x⁶ - 27

This is already as simplified as it can be.

Which of the following intervals contains all the solutions to the equation: log₅ (𝑥²+1) = log₅ (𝑥) + log₅(2)

[0.5,3.1]

Which of the following intervals has the solution to the equation: (1/256)⁻³ˣ⁺⁴ = (16)⁴ˣ⁺¹⁰

[3.7,5.5]

Which of the following intervals contains all the solutions to the equation: log₂ (𝑥+1) − log₂ (5)= log₂(𝑥)

[−0.5,1.5]

Which of the following intervals has the solution to the equation:(¹⁄₈₁)ˣ = (⅓)⁻³ˣ⁺¹

[−1.2,0.5]

Identify the coordinates of the points shown on the graph.

a: (0,4); b: (2,1); c: (-5, -3)

Which of the following is equivalent to the expression : e²ˣ² · eˣ / √e

e²ˣ²⁺ˣ⁻ 1/2

Consider the following graph: Which of the following could be the function that generated the above graph? (Hint: Compare points on the graph to points of the function)

f (x) = [line 1: -x² ~ x ≤ 2 [line 2: 1/3 (x-1)³ - 3 ~ 2 < x ≤ 4 [line 3: √x + 5 - 1 ~ 4 < x

Which of the following is an inverse of the function 𝑓(x)=x³+ 3x²+3𝑥+1 with the largest valid domain of f(x) that is inverted.

f(x) is inverted on the domain all real numbers, and the inverse function is f⁻¹ (y) = ³√y-1

Which of the following is an inverse of the function f (x) = x² + 4x + 18 with the largest valid domain of 𝑓(𝑥) that is inverted

f(x) is inverted on the domain x ≥ -2, and the inverse function is f⁻¹ (y) = y - 14 - 2. FIGURE*

Which of the following graphs would most properly be said to have the parent function 𝑓(𝑥)= 𝑒ˣ

the graph slowly going up in one direction.

Which of the following graphs depicts a continuous relationship?

the one that has no breaks. or empty. no holes.

Which of the following is the graph of 𝑓(𝑥)=(−𝑥+2)³ -1

the one where the arrow touches directly under 5. and hits -10

If a < b must it be true that ³√𝑎 < ³√𝑏

true

Consider the following rational function: 𝑟(𝑥)= 𝑥³−12𝑥+16/ 𝑥²−5𝑥+6 Which of the following are vertical asymptotes of ? (Select all that apply

x = 3

What values of x satisfy the equation: √10x-15 = √8x-16 +1

x = 4

Consider the following rational function: 𝑟(𝑥)= x⁴ + 7x³ +18x² +20x + 8 / x² - 3x - 10 Which of the following are vertical asymptotes of ? (Select all that apply)

x = 5

What value(s) of 𝑥 satisfy the equation: √9x - 9/4 = √5x - 25/4 + 2

x= 5/2

Which value of 𝑥 satisfies the equation: ln (𝑥−1) + ln (3) = ln(𝑥)

x= ³⁄₂

Which value(s) of x satisfy the equation 25³ˣ ⋅ (⅕)³ˣ⁺¹

x= ⁻¹⁄₃

What is the result when dividing 2x⁵ −13x⁴ +11x⁴ +8x² +5x−12 / 2x−3

x⁴− 5x³ −2x² +x+4

Which of the following inequalities best models the phrase "x is at least 12 units from y''?

| 𝑥 − 𝑦| ≥ 12

Compute the following numeric exponential: (⅔)⁻²

⁹⁄₄

Consider the polynomial p(x)=45𝑥+69𝑥²−24𝑥−48. What are the (real-valued) zeros of 𝑝(𝑥)

⁻⁴⁄₃, -1, ⅘

Suppose f(x) has a zero at an x-value of 3. What would a zero of g(x) be if 𝑔(𝑥) = −2𝑓 (3𝑥 + 1)

⅔

Factor the radicand of the following radical √ 2x²+15x+28

√(2𝑥+7)(𝑥+4)

Before simplifying the radical √3x³ -25x² + 24x + 28, one should convert it to which of the following?

√(3x + 2) (x - 2) (x - 7)

Which of the following is best described as a mathematical expression?

√a²+b²

Which of the following is equivalent to the expression: e^1/2ˣ^²⁺³ˣ⁻¹

√eˣ^²(ᵉˣ)³ / e

You want to study the replication of a particular strain of E. coli bacteria and have the following information: The particular strain of E. coli you want to study doubles in population after 5 hours under ideal (i.e. laboratory) conditions. Most strains of E. coli double in population after 7 hours under ideal conditions. Use of the laboratory space and equipment costs $80 per hour. You need to observe at least 8 generations of the E. coli. You have a grant in the amount of $3,500 to fund your study. Given that you begin with an initial population of 100 bacteria, which of the following functions models the size of the E. coli population after 𝑡t hours?

𝑓(𝑡) = 100⋅ 2 ^ᵗ⁄₅

Which of the following is the piecewise definition of the function 𝑓(𝑥) = | 3x + 1 |

𝑓(𝑥)= [ −3𝑥−1 𝑥 ≤ ⁻¹⁄₃ [3x + 1 ⁻¹⁄₃ <𝑥

Consider the following graph: starts with squiggly going up with open dot. second has two closed dot going down from 2-5. third one starts with open circle, ends with arrow going straight up from 5-10. Which of the following could be the function that generated the above graph? (Hint: Compare points on the graph to points of the function)

𝑓(𝑥)= [½𝑥³ +4, 𝑥 < 2 [-2√𝑥−1 +5, 2 ≤ 𝑥 ≤5 [ 2^ ⅕ x-1 +3, 5 < 𝑥

Consider the polynomial 𝑝(𝑥)= −2𝑥⁹ +4𝑥⁸ +3𝑥⁴ −2𝑥³ −𝑥² . What can be said about the extrema of 𝑝(𝑥)?

𝑝(𝑥) does not have an absolute extrema and it has at most 8 local extrema.

The function 𝑓 (𝑟) = 𝑟²+ 2𝑟 + 𝐶 calculates the cost of food, given a certain number of rabbits; with a missing constant 𝐶 . What is the independent variable in this function?

𝑟

Find all real values of 𝑥 that satisfy the equality: |6𝑥²−4𝑥| = 10

𝑥= ⁵⁄₃,−1

If f(x) = 𝑥² + x − 1 and g(x) = x − 1. What is (g∘f)(𝑥)

𝑥² + 𝑥 - 2

If f(x) = x² +𝑥 − 1 and g(x) = x − 1. What is (f−2g)(x)?

𝑥² - 𝑥 + 1

For the following graph which is the most likely parent function? wavy s across graph

𝑥³

What is the parent function of 𝑔(𝑥)= 𝑥² −3𝑥³ + 𝑥−2?

𝑥³

Consider the following rational function: 𝑟(𝑥)= 𝑥² + 8𝑥 + 15 / 𝑥² + 2𝑥 - 15 Which of the following are domain restrictions of r(x)? (Select all that apply)

𝑥≠ 3 𝑥≠ − 5

Consider the following rational function: 𝑟(𝑥)= x²+x-2 / x² -3x-10 Which of the following are domain restrictions of r(x)? (Select all that apply)

𝑥≠5, 𝑥≠−2

Consider the following rational function: 𝑟(𝑥)= 𝑥²+2𝑥−15 / 𝑥²+9𝑥+20 Which of the following are domain restrictions of r(x)? (Select all that apply)

𝑥≠−4 𝑥≠−5

Which of the following is equivalent to the logarithm: log₂ (8x³ / x²-1)

3+ 3 log₂ (𝑥) − log₂(𝑥−1) − log₂(𝑥+1)

Compute the following numeric exponential: (½)⁻⁵

32

A bacterium triples in size every 6 minutes. If you start with 3 bacterium in a dish, which of the following calculates the total amount of bacteria after t hours?

3¹⁰ᵗ⁺¹

You take out a personal loan for $4000 with an interest rate of 4% a month, compounded four times a month. What is the model to determine how much you owe if you don't make any payments, after 𝑡t months?

4000 (1 +¹⁄₁₀₀) ⁴ᵗ

Which of the following is equivalent to √375x⁶ (x+2)³

5 |x³(𝑥+2)| √15 (x+2)

Simplify the numeric radical ³√2160.

6³√10

Simplify the numeric radical √2695

7 √55

Consider the polynomial 𝑝(𝑥)= 𝑥⁴ − 4𝑥³ −47𝑥²+210𝑥 Which of the following are true? I: p(x) is degree 4. II: p(x) has 4 real zeros. III: The sum of the zeros of p(x) is 4. IV: The product of the zeros of p(x) is 0

All of these are correct.

Consider the following graph of f(x) graph has letters a-e. looks like roller coaster. Within which of the following segments is the function increasing and concave down?

Between b and c

At the Friends of the Library book sale, the books are arranged into a number of categories based on their value as follows; Tier 1: $2.50 Tier 2: $5.00 Tier 3: $10.00 You would like to plan your spending at the booksale before your arrival; so you still have enough money to buy food for the upcoming week. You decide the easiest way to do this is to write an equation for how much you spend when you buy 𝑥 Tier 1 books, 𝑦 Tier 2 books and 𝑧 Tier 3 books. Which of the following gives the total cost (in dollars) for buying books at the sale given the previous variable assignments?

C = 2.5x+5y+10z

What is the primary difference between even and odd roots? Select all of the below that are true.

- Even roots must use restricted domains to get a partial inverse of an even power which leads to a host of nuance issues like absolute values and domain problems. -Odd roots are inverses to potentially one to one functions which means it can have a domain of all real numbers and doesn't need absolute values.

Which of the following are defining characteristics of exponential growth or decay; as oppose to other forms of increase or decrease. (Select all that apply)

- The ratio between any two consecutive iterations should be approximately the same value. - The growth/decay depends on the amount there currently is.

What is the purpose of using Piece-wise functions? (Select all that apply)

- To stitch together several types of functions on separate domains to make one comprehensive graph. - For use when the underlying model changes at different points of a domain.

Consider the function F(X) = [−2𝑥−8 −2≤𝑥≤2 [ −5√ |𝑥 +1 | -5. 2<𝑥≤6 [−4(𝑥−1)2+2 6<𝑥≤12 What is f(2)?

-12

Consider the function 𝑓(𝑥)= [5(𝑥+1)³+5, −1≤𝑥≤3 [−4𝑥+3, 3<𝑥≤6 [𝑥−1, 6<𝑥≤7 What is f(4)?

-13

What is the remainder when −4𝑥⁴ +𝑥³ +11𝑥² −3𝑥−1 is divided by −𝑥−3?

-244

Consider the polynomial 𝑝(𝑥)=4𝑥³ +16𝑥² −25𝑥 −100. What is the sum of the zeros of p(x)?-

-4

Consider the polynomial p(x)= 45x³ + 69x² −24x − 48. What are the zeros of 3𝑝 (𝑥 + 1)

-7/3, -2, -1/5

What are notable features of an exponential graph? (Select all that apply)

-A horizontal asymptote -A rapid vertical change after some kind of turning point.

Which of the following would be likely situations to use exponential growth or decay type models? (Select all that apply)

-Compound interest -Amount of radioactive material remaining over time. -Bacterial growth

What kind of things would one use absolute value for? (Select all that apply)

-Distances -Forcing an output to be positive (eg results of even roots)

Which of the following are key traits of a logarithmic graph? (Select all that apply)

-There is a vertical asymptote -The domain is open on one side (meaning that the domain has a parenthesis instead of a bracket)

A function that determines what a house costs (in dollars), dependent on it's square footage has a domain of:___and a codomain of____.

1.square footage of houses 2.money (in dollars)

Compute the following numeric exponential: 7⁻³

1/343

Which of the following is equivalent to the expression: ¹⁄₁₀₀ 10ˣ 10³ ⁻ ˣ / 100ˣ²⁻¹

10³⁻²ˣ^²

Which of the following is equivalent to the expression 10^-3x²+x-1

10ˣ / 10x1000ˣ²

Your credit card company is changing their rates and sends you a promotional card to let you know the new terms. Currently you have $1200 on your credit card, and due to the change over promotion you don't need to make any payments for 6 months. The new interest rate is 8% compounded weekly. If you don't make any payments in the 6 month (26 week) promotional period, how much would you owe on your credit card?

1200 (1 + ¹⁄₆₅₀)²⁶

Consider the polynomial 𝑝 (x)= −9x⁸ −5x⁸ −4x¹³- 3x. How many zeros does this polynomial have including complex valued zeros and counting multiplicity?

13

Which real value(s) of x satisfy the equation √4x+3 =2 √x-1 +1

13/4

You have a student loan that accrues interest at a 12% APR (annual percentage rate). If the interest is compounded weekly, what is the growth/decay multiplier in the exponential model of how much you owe on your loan?

1303/1300

Which of the following is equivalent to the expression: 2²ˣ^²⁻³ˣ⁺¹

2 · 4ˣ² / 8ˣ

Use the properties of logarithms to simplify: log₂ (16xy³) - log₂ (4xy)

2+ 2 log₂ (y)

Which of the following is equivalent to log₇(25)

2/log₅ 7

Fully factor the following polynomial: 25𝑥⁴−50𝑥²+25

25(𝑥−1)² (𝑥+1)²

Which of the following is the fully simplified form of³√120x⁴ (x + 2)⁵?

2x (x + 2) ³√15x(x + 2)²

Full factor the polynomial 6x⁵−8x⁴−54x³+72x²

2x² (3x-4)(x-3)(x+3)

Which of the following is equivalent to the expression: 4·2ˣ²⁺¹·2ˣ / √2·2ˣ⁺¹

2ˣ²⁺¹.⁵

Full factor the polynomial 6𝑥⁵−8𝑥⁴−54𝑥³+72𝑥²

2𝑥²(3𝑥−4)(𝑥−3)(𝑥+3)

Which of the following is equivalent to the logarithm ln (e³ ³√𝑥 /𝑥𝑦)

3 - ⅔ 1n (x) - 1n(y)

Which of the following is equivalent to the logarithm 1n(e³ ³√x / xy)

3 - ⅔ 1n (x) - 1n(y)

Fully factor the polynomial: 24𝑥³ −8𝑥² −96𝑥 + 32

8 (x-2)(x+2)(3x-1)

Consider the function 𝑓(𝑥)= [3𝑥² +5, 𝑥 ≤ −4 [−2𝑥, −4< 𝑥 <3 [𝑥²+3, 3 ≤ 𝑥 What is f(9)?

84

Fully factor the following polynomial: 9𝑥³ − 72

9 (x - 2) (x² + 2x + 4)

Simplify the numeric radical √486

9 √6

Consider the following rational function: 𝑟(𝑥)= 𝑥⁴+7𝑥³+18𝑥²+20𝑥+8 / 𝑥²−3𝑥−10

There is no Horizontal Asymptote

Which of the following is equivalent to the logarithm: log (¹⁄₁₀₀ (x²+1) / x⁴ (x+1)

log (x²+1) - 4log (x) - log (x+1) -2

Which of the following is the most condensed form of the following logarithmic expression: log₂(x²) - 3log₂ (x+1) + log₄ (3x)

log₂ (√3x^⁵⁄₂ / (x+1)³

Which of the following is the most condensed form of the following logarithmic expression: 2log₃(3𝑥)− log₃ (2𝑥)+ log₃ (𝑥²+1)

log₃ (9x(x²+1) / 2

Which of the following is equivalent to log₃ (250)

log₅ 250 / log₅ 3