practice final
Consider the function f(x)={5(x+1)3+5−1≤x≤3−4x+33<x≤6x−16<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
Translate the following expression into a mathematical expression: "The distance between 24x3−42y2 and z is no more than 40".
|24x^3 -42y^2-z| <- 40
What is the domain of function: f(x)=−ln(x+1)+12
(-1, infinity)
What is the domain of function: f(x)=−3ln(−2x−14)+1
(-infinity, -7)
What is the domain of the real-valued function f(x)=−13−3x+17 - 1
(-infinity, 17/3)
Fully factor the following polynomial: 3x3−4x2−192x+256
(x+8)(3x-4)(x-8)
Fully factor the following polynomial: x2−22x+121
(x-11)^2
7^-3
1/343
Compute the following numeric exponential: (1/2)^-5
32
For each of the following questions, consider the function A:P→Y where P is the set {Jake, Ashley, Elijah, Maria}, Y is the set of positive whole numbers, and the function is given by the rule laid out in the following table:
Y
Which of the following intervals contains all the solutions to the equation: log5(x2+1)=log5(x)+log5(2)
[0.5, 3.1]
What kind of things would one use absolute value for? (Select all that apply)
distances forcing an output to be positive
Which of the following are key traits of a logarithmic graph? (Select all that apply)
domain is open on one side- meaning domain has a parenthesis instead of a bracket there is a vertical asymptote
If 0<a<1, is there an integer n such that n square root a≥1?
no
Which of the following could be the function that generated the above graph? (Hint: Compare points on the graph to points of the function)
x <- 2 2< x <- 4 4< x
Which of the following denotes a point that is left 2 and up 2 from the origin?
(-2,2)
Fully factor the following polynomial: 81x^3−18x^2−9x+2
(3x+1)(9x-2)(3x-1)
Which of the following is equivalent to the expression square root of 125x3y6− square root of 5x?
(5|xy^3| -1) square root of 5x
Factor the following quadratic: 30x^2+59x+28
(6x+7)(5x+4)
Factor the following polynomial: 64x^2−121
(8x+11)(8x-11)
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
Solve the following absolute value equation. |3x+1|=2x+14How many (real) solutions does this equation have?
2
How do we use the Cartesian Plane in this class?
TO RELATE VARIABLES
A polynomial has an absolute maximum when:
The leading term has an even exponent and the leading coefficient is negative.
Which of the following intervals contains all the solutions to the equation: log2(x+1)−log2(5)=log2(x)
[-0.5, 1.5]
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 e^2x2⋅ex/ square root e?
e^ (2x^2 +x- 1/2)
Which of the following is equivalent to the expression e^2x^2⋅ex^e/ square root e
e^ 2x^2+ x - 1/2
Over what interval is the following graph concave up? ( [ Select ] , [ Select ] ) (NOTE: put N/A for both answers if there are no intervals that match the requirement)
n/a n/a
Consider a function f which maps each letter of the alphabet to a positive integer. The function takes each letter and maps it to its opposite" position in the alphabet. For example, f(A)=26, f(B)=25, f(Y)=2, and f(Z)=1. What is the codomain of the function f?
set of all positive integers
Factor the following radicand to force the type two radical into being a type one radical: square root 2x^3−x^2−4x+3
square root (2x^3- x^2-4x+3) = square root (x-1)^2(2x+3)
Before simplifying the radical 3x3−25x2+24x+28, one should convert it to which of the following?
square root (3x+2) (x-2)(x-7)
Consider a function f which maps each letter of the alphabet to a positive integer. The function takes each letter and maps it to its opposite" position in the alphabet. For example, f(A)=26, f(B)=25, f(Y)=2, and f(Z)=1. What is the domain of the function f?
the set of all letters of the alphabet
Suppose you are running ballistics for the US Army, and you have a function which plots a projectile's height over time. What would a zero of this function represent?
the time when the projectile hits the ground
simplify the expression 4 square root (x^4-16)
this is already simplified as it can be
Simplify the expression 3 square root x^6−27
this is simplified
Consider the following rational function: r(x)=x2+x−2x2−3x−10 Which of the following are domain restrictions of r(x)? (Select all that apply)
x does not equal -2 x does not equal 5
Which of the following is the piecewise definition of the function f(x)=|3x+1| Group of answer choicesf(x)={−3x−1x≤133x+113<x
x<- -1/3 -1/3< x
Which of the following could be the function that generated the above graph? (Hint: Compare points on the graph to points of the function)
x<2 2<- x <- 5 5<x
If f(x)=x^2+x−1 and g(x)=x−1. What is (f−2g)(x)
x^2-x+1
What is an expression for |x−3| when x<3?
3-x
Consider the polynomial p(x)=4x3+16x2−25x−100. What is the sum of the zeros of p(x)?
-4
Evaluate the following logarithm without using a calculator. log27(3)
1/3
Sam put 10,000 dollars into an account 15 years ago. Their account is worth 28,489.47 dollars today. Given that the account compounds monthly, which of the following represents the interest rate (in percent) on the account? Group of answer choices1200(2.848947180−1)
1200 (180 square root of 2.848947 ) - 1
Use the properties of logarithms to simplify: log2(16xy3)−log2(4xy)
2+2log2(y)
Which of the following is equivalent to log7(25)? Hint: Use change of base formula.
2/log5 (7)
Which of the following is the fully simplified form of 3 square root (120x^4(x+2)^5)
2x(x+2) 3 square root (15x(x+2)^2
Consider the function f(x)={3x2+5x≤−4−2x−4<x<3x2+33≤x What is f(9)?
84
Simplify the numeric radical 486
9 square root 6
Which of the following is the most condensed form of the following logarithmic expression: 2log3(3x)−log3(2x)+log3(x2+1)
log3 (9x (x^2+1) /2)
Say you're watching Florida play a close basketball game against Kentucky, and they display the graph of a function which plots the difference between the scores of the 2 teams over time. What would the zeros of this graph represent?
The point(s) in time when the two teams are tied
Consider the function: f(x)={x+7−2≤x<42x−54≤x≤8 Which of the following intervals contains the value for f(−1)?
[2,9]
Which coordinates describe a point that is 5 units up and 2 units to the left of the origin?
(-2,5)
Consider the following rational function: r(x)=x2+2x−15x2+9x+20 If the function fails to display properly the function is: r(x) = [ x^2+2x-15] / [ x^2+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 (ie no non-real numbers allowed) function f(x)=2−x+1+7?
(-infinity, 1]
What is the domain of function: f(x)=log2(−3x+6)−1
(-infinity, 2)
onsider the following rational function: r(x)=x4−x3−13x2+25x−12(x−1)2 If it fails to display properly the function is: r(x) = [ x^4-x^3-13x^2+25x-12 ] / [ (x-1\right)^2 ] Which of the following are coordinates of holes on the graph of r(x)? (Select all that apply)
(1,-10)
What is the fully factored form (with real coefficients) for the polynomial f(x)=x4−256? (Hint: 4^4=256)
(x-4)(x+4)(x^2+16)
Over what interval is the following graph decreasing? ( [ Select ] , [ Select ] ) (NOTE: put N/A for both answers if there are no intervals that match the requirement)
-infinity, 0
OVER WHAT INTERVAL IS THE FOLLOWING GRAPH INCREASING
-infinity, infinity
Over what interval is the following graph concave down? ( [ Select ] , [ Select ] ) (NOTE: put N/A for both answers if there are no intervals that match the requirement)
-infinity, infinity
Solve the following absolute value equation. |x−4|=−8How many (real) solutions does this equation have?
0
How many "jump" discontinuities are depicted in the following graph?
1
Which of the following is equivalent to the expression: 1/100 10^x 10^3-x/ 100^x^2 -1
10^3-2x^2
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
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
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
How many "infinite" aka "asymptotic" discontinuities are depicted in the following graph?
2
Which of the following is equivalent to the expression: 2^2x2−3x+1
2*4^x2/ 8^x
Full factor the polynomial 6x5−8x4−54x3+72x2
2x^2 (3x-4) (x-3) (x+3)
Consider the function f(x)=5x3+2+x−3x3. How many zeros does f(x) have including complex valued zeros and counting multiplicity?
3
Consider the polynomial p(x)=45x3+69x2−24x−48 How many zeros does this polynomial have counting complex value zeros and multiplicity?
3
Evaluate the following logarithm without using a calculator. log4(8)
3/2
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?
3^10t+1
What is the remainder when dividing (3x4−x3+8x2+8x+7) by (x2+x+1) ?
3x-2
How many local extrema are depicted in the following graph?
4
How many local extrema are depicted in the following graph?
5
Simplify the numeric radical 5 square root of 3125
5
Simplify the following expression using rules for exponents so that the result only has one exponential. 5a+2⋅25−b−4252a−3⋅25b+1
5^-3a -4b-2
What is the lowest possible degree of the function represented by the following graph?
6
Consider the function f(x)=x7+3x2+x+1. How many zeros does f(x) have including complex valued zeros and counting multiplicity
7
What is the lowest possible degree of the function represented by the following graph?
7
Simplify the numeric radical square root 2695
7 square root 55
Simplify the numeric radical square root 486
9 square root 6
What is the Cartesian Plane?
A coordinate axis system measuring the x distance and y distance.
Consider the polynomial f(x)=x2+3x+2. Which of the following statements aretrue? I: f(x) has 2 roots but no real zeros.II: f(x) has 2 roots and 2 real zeros.III: f(x) is degree 2 which is why it must have 2 real zeros.IV: f(x) is degree 2 but has no real zeros.
ONLY 2 IS CORRECT
Which of the following intervals has the solution to the equation: (1/256)^−3x+4=(16)^4x+2
[3.7, 5.5]
Which of the following intervals contains all the solutions to the equation: square root x−4+ square root x−9=5
[8,19]
Which of the following intervals contains all the solutions to the equation: x−4+x−9=5
[8,19]
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 t hours?
f(t) = 100 x 2 ^ t/5
Consider the function f(x)={−2x−8−2≤x≤2−5|x+1|−52<x≤6−4(x−1)2+26<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!
jump discontinuity
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
Is the Cartesian Plane only used in mathematics?
no, used in non-academic areas
Which values of x satisfy the following equation? square root x+1+square root x+7=−3
none of these are correct
What is the best parent function of the function shown on the following graph?
px= e^x
What is the best parent function of the function shown on the following graph?
px= x^2
Before simplifying the radical square root 3x^3−25x^2+24x+28,
square root (3x+2)(x-2)(x-7)
Which of the following is equivalent to the expression: e^1/2x^2+3x−1
square root ex^2 (e^x)^3/ e
Say you are working as an economist for a large company, and you develop a function which gives the estimated net profit when given a certain amount of man-hours invested into Research and Development. What would a zero of this function represent?
the amount of man hours you need to invest to break even
Consider the function f(x)={−2x−8−2≤x≤23|x−1|−152<x≤6−4(x−1)2+26<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
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 growth/decay depends on the amount there currently is the ratio between any two consecutive iterations should be approximately the same value
Consider the following rational function: r(x)=7x3+18x2+20x+82x4+x2−3x−10
the horizontal asymptote is y=0
Consider the following rational function: r(x)=x4+7x3+18x2+20x+8x2−3x−10
there is no horizontal asymptote
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 following rational function: r(x)=x3−12x+16x2−5x+6 Which of the following are vertical asymptotes of ? (Select all that apply)
x=3
Consider the following rational function: r(x)=x3+6x2−32x2−16 Which of the following are vertical asymptotes of ? (Select all that apply)
x=4
If f(x)=x^2+x−1 and g(x)=x−1. What is (g∘f)(x)?
x^2+ x-2
What is the result when dividing 2x5−13x4+11x3+8x2+5x−12 by 2x−3?
x^4-5x^3-2x^2+x+4
can the expression |x| -1 ever be negative
yes
Translate the expression into mathematical expressions. "The distance between ab2+bc2+ca2 and ab2c+a2bc3 is smaller than 1"
| ab^2 + bc^2 +ca^2- ab^2c- a^2bc^3| < 1
Translate the expression into mathematical expressions. "The distance between xyz and xy−yz is at least 122"
|xyz-xy+yz| >- 122
If a<b must it be true that a^3<b^3 ?
true
What is the domain of the real-valued function f(x)=−13 (6 square root of )−3x+17 −1
(-infinity, 17/3]
Factor the following quadratic: 36x^2−59x−7 This is already fully factored
(9x+1)(4x-7)
Simplify the following expression as much as possible. log5(5a+2⋅25−b−4252a−3⋅25b+1)
-3a-4b-2
Consider the polynomial p(x)=45x^3+69x^2−24x−48 What are the zeros of 3p(x+1)?
-7/3, -2, -1/5
What is the remainder when dividing (x3−2x2−17x+20) by (x+4)?
-8
Expand the following logarithm into the sum of simplified logs:log((a2⋅b5⋅c6a−1⋅b7⋅c4)−3)
-9log(a) + 6log (b)- 6log(c)
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.
Consider the polynomial p(x)=−13x^12−4x^19+3x^9+2x How many zeros does this polynomial have including complex valued zeros and counting multiplicity?
19
What is the remainder when dividing (x^4+5x^3+6x^2+26x−17) by (x+5)?
3
how many "hole" discontinuities are depicted in the following graph?
3
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
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.
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? Group of answer choicesC = 2.5x+5y+10z
C= 2.5X + 5Y+ 10Z
For each of the following questions, consider the function A:P→Y where P is the set {Jake, Ashley, Elijah, Maria}, Y is the set of positive whole numbers, and the function is given by the rule laid out in the following table: xA(x)Jake4Ashley6Elijah5771Maria5 What is the domain of this function?
P
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. What would be the independent variable in the generalized relationship that determines your potential parking spaces based on your permit color?
THE COLOR OF YOUR PERMIT
A polynomial has an absolute minimum when:
The leading term has an even exponent and the leading coefficient is positive.
Say you are working as an economist for a large company, and you develop a function which gives the estimated net profit when given a certain amount of man-hours invested into Research and Development. What would a local maxima of this function represent?
The number of man-hours, along with the corresponding net profit, which corresponds to highest possible profit
Which of the following would be likely situations to use exponential growth or decay type models? (Select all that apply)
amount of radioactive material remaining over time compound interests bacterial growth
A function that determines what a house costs (in dollars), dependent on it's square footage has a domain of: [ Select ] and a codomain of [ Select ] ?
domain: square footage of houses codomain: money in dollars
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.
is inverted on the domain x≥−2, and the inverse function is f−1(y)=square root of y−14−2
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
