Answer:
[tex]f^{-1}(x)=\dfrac{1}{3}(\ln x +1)[/tex]
Step-by-step explanation:
Given function:
[tex]f(x)=e^{3x-1}[/tex]
Replace f(x) with y:
[tex]\implies y=e^{3x-1}[/tex]
Take natural logs of both sides:
[tex]\implies \ln y = \ln e^{3x-1}[/tex]
Apply the Power Log Law [tex]\ln x^n=n\ln x[/tex] :
[tex]\implies \ln y = (3x-1)\ln e[/tex]
As [tex]\ln e=1[/tex] then:
[tex]\implies \ln y = 3x-1[/tex]
Rearrange to make x the subject:
[tex]\implies 3x=\ln y +1[/tex]
[tex]\implies x=\dfrac{1}{3}(\ln y +1)[/tex]
Swap x for [tex]f^{-1}(x)[/tex] and y for x:
[tex]\implies f^{-1}(x)=\dfrac{1}{3}(\ln x +1)[/tex]
City there are 244,000 48-year-olds. Based on the table below how
many are not expected to be alive in a year?
Answer:
405
Step-by-step explanation:
It tells in the table.
Reflectional Symmetry
Instructions: If the following figures have
reflectional symmetry, determine the
number of lines of symmetry.
Lines of Symmetry:
The figure has a rotational symmetry and the number of lines of symmetry is 3
How to determine the rotational symmetry?The given figure is an equilateral triangle.
An equilateral triangle has 3 equal sides and 3 equal angles.
This means that it has a rotational symmetry and the number of lines of symmetry is 3.
The angle of rotational symmetry is calculated as:
Angle = 180/3
Evaluate
Angle = 60
Hence, the angle of rotational symmetry is 60
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Consider the graph of f(x) = (x + 5)2 – 3. Rewrite the equation by making one change in the equation of f(x) so that the graph has no x-intercepts.
Using the discriminant of a quadratic equation, if the graph is translated shifted up 4 units, the graph will have no x-intercepts.
What is the discriminant of a quadratic equation and how does it influence the solutions?A quadratic equation is modeled by:
[tex]y = ax^2 + bx + c[/tex]
The discriminant is:
[tex]\Delta = b^2 - 4ac[/tex]
The solutions are as follows:
If [tex]\mathbf{\Delta > 0}[/tex], and it has 2 x-intercepts.If [tex]\mathbf{\Delta = 0}[/tex], it has 1 x-intercept.If [tex]\mathbf{\Delta < 0}[/tex], it has no x-intercepts.In this problem, the function is given by:
f(x) = (x + 5)² - 3.
In standard form:
f(x) = x² + 10x + 22.
We want to find coefficient k for which the function has [tex]\Delta < 0[/tex], then:
f(x) = x² + 10x + 22 + k.
The coefficients are a = 1, b = 10, c = 22 + k, hence:
[tex]\Delta < 0[/tex]
10² - 4(22 + k) < 0
100 - 88 - 4k < 0
4k > 12
k > 3.
Hence, with k = 4, the function is shifted up 4 units, and the graph will have no x-intercepts.
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Verify that (cos²a) (2 + tan² a) = 2 - sin² a....
Trigonometric Formula's:
[tex]\boxed{\sf \ \sf \sin^2 \theta + \cos^2 \theta = 1}[/tex]
[tex]\boxed{ \sf tan\theta = \frac{sin\theta}{cos\theta} }[/tex]
Given to verify the following:
[tex]\bf (cos^2a) (2 + tan^2 a) = 2 - sin^2 a[/tex]
[tex]\texttt{\underline{rewrite the equation}:}[/tex]
[tex]\rightarrow \sf (cos^2a) (2 + \dfrac{sin^2 a}{cos^2 a} )[/tex]
[tex]\texttt{\underline{apply distributive method}:}[/tex]
[tex]\rightarrow \sf 2 (cos^2a) + (\dfrac{sin^2 a}{cos^2 a} ) (cos^2a)[/tex]
[tex]\texttt{\underline{simplify the following}:}[/tex]
[tex]\rightarrow \sf 2cos^2 a + sin^2 a[/tex]
[tex]\texttt{\underline{rewrite the equation}:}[/tex]
[tex]\rightarrow \sf 2(1 - sin^2a ) + sin^2 a[/tex]
[tex]\texttt{\underline{distribute inside the parenthesis}:}[/tex]
[tex]\rightarrow \sf 2 - 2sin^2a + sin^2 a[/tex]
[tex]\texttt{\underline{simplify the following}} :[/tex]
[tex]\rightarrow \sf 2 - sin^2a[/tex]
Hence, verified the trigonometric identity.
Answer:
See below ~
Step-by-step explanation:
Identities used :
⇒ cos²a = 1 - sin²a
⇒ tan²a = sin²a / cos²a
============================================================
Solving :
⇒ (cos²a) (2 + tan² a)
⇒ 2cos²a + (cos²a)(tan²a)
⇒ 2(1 - sin²a) + sin²a
⇒ 2 - 2sin²a + sin²a
⇒ 2 - sin²a [∴ Proved √]
PLEASE HELP!!! DSFVAGETRYHUJ
Answer: Remeber 2+2=5 and
Step-by-step explanation: if yur on fire stop drop and roll do it and your awesome. Another thing is if someones taking forever drinking water when your behind them sing 123 can you hurry up please 456 imma hit you with a brick 789 imma spit this sick ryme
The Father of the Bride flips out when he realizes that the hot dogs and buns don't
match. There are 8 hot dogs in a package and 12 buns in a package. How many
packs of hot dogs and how many packs of buns do you need to purchase in order to
have the exact same number of each?
Using the least common multiple, it is found that you need to buy 3 packs of hot dogs and 2 packs of buns to have the exact same number of each.
How to find the number of packs needed?First, we need to find the total number of each element, finding the least common multiple of the sizes of each package, that is, 8 and 12.
The lcm is found factoring them by prime factors, hence:
8 - 12|2
4 - 6|2
2 - 3|2
1 - 3|3
1 - 1
Hence, lcm(8,12) = 2³ x 3 = 24.
Then:
24/8 = 3.24/12 = 2.Which means that you need to buy 3 packs of hot dogs and 2 packs of buns to have the exact same number of each.
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The following exercises present you with fairly straightforward ratio and proportion situations.
On one map, -inch represents 18 miles. If two cities are inches apart on the map, what is the actual distance (in miles)
between them?
Answer:
36+ Miles
Step-by-step explanation:
One in being 18 miles, and two cities that are inches apart. So 2 or more inches. Would make it anywhere from 36 or more miles.
A point that is exterior to an angle is found on the acute side of the angle. true of false?
A point that is exterior to an angle is found on the acute side of the angle which is true.
What is an angle?The angle is the distance between the intersecting lines or surfaces. The angle is also expressed in degrees. The angle is 360 degrees for one complete spin.
We know that the sum of the exterior angle and interior angle is equal to 180 degrees.
"If one of the exterior angles is an acute angle, then the corresponding interior angle is an obtuse angle."
A point that is exterior to an angle is found on the acute side of the angle which is true.
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Arun took a taxi from his house to the airport. The taxi company charged a pick-up fee of $4 plus $4.25 per mile. The total fare was $110.25, not including the tip. How many miles was the taxi ride?
Here we must write and solve a linear equation to find the number of miles that Arun traveled in the taxi. We will find that Eva traveled 11 miles.
So we know that the taxi charges a fee of $4.10 and then a plus of $0.50 per mile.
So if you travel for m miles, the cost equation is:
C(m) = $4.10 + $0.50*m
Now, we know that for Eva the total fare (total cost) was $9.60, then we need to solve:
$9.60 = C(m) = $4.10 + $0.50*m
$9.60 = $4.10 + $0.50*m
$9.60 - $4.10 = $0.50*m
$5.50 = $0.50*m
$5.50/$0.50 = m = 11
This means that Arun traveled 11 miles in the taxi.
The Eva traveled 25 miles in the taxi.
It is required to find how many miles was the taxi ride.
What is arithmetic?Arithmetic is the branch of mathematics that deals with the study of numbers using various operations on them. Basic operations of math are addition, subtraction, multiplication and division. These operations are denoted by the given symbols.
Given:
The taxi company charged a pick-up fee of $4 plus $4.25 per mile.
The total fare was $110.25.
The cost equation is:
C(m) = $4 + $4.25*m
Now, we know that for Eva the total fare (total cost) was $110.25, then we need to solve:
$110.25 = C(m) = $4 + $4.25*m
$110.25-$4=$4.25*m
$106.25=$4.25*m
m =25
Therefore, the Eva traveled 25 miles in the taxi.
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Find the area of a regular
polygon with 8 sides that has a
the side length of 4 inches and an
apothem of 5 inches.
Area = [?]in2
Answer:
80 in²
Step-by-step explanation:
When side length and apothem of a regular polygon are given, the applicable area formula is ...
A = 1/2Pa
where P is the perimeter of the polygon, and 'a' is its apothem.
__
The perimeter of a regular 8-sided polygon will be 8 times the side length, so the area is ...
A = 1/2(8×4 in)(5 in) = 80 in²
Select from the drop-down menus to correctly complete the sentence. The probability of a chance event is a number between 0 and 1 that expresses the __?___ that an event __?___ occur.
The probability of a chance event is a number between 0 and 1 that expresses the likelihood that an event would occur.
How to complete the blanks?As a general rule in probability, we have:
Minimum probability = 0Maximum probability = 1The above represents the range of a valid probability (both inclusive).
This also represents the likelihood that an event would occur
Hence, the words that complete the blank are likelihood and would.
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Which statements describe one of the transformations performed on f(x) = x^2 to create g(x) = (x - 2)^2 + 5? Choose all that apply.
A. A translation of 5 units up
B. A translation of 2 units to the right
C. A translation of 2 units to the left
D. A translation of 5 units down
n⃗ =〈−2,−1〉 and D=[−4 4 2 3].
What is D⋅n⃗ ?
Enter your answer as a vector by filling in the boxes.
The dot product of the two matrices D and n is determine as (4, - 7).
Dot product of the vector
The dot product of the two matrices is calculated as follows;
n = (-2, -1), and D = [-4 2]
[4 3]
D.n = (-2, -1). [-4 2] = (-2 x -4) + (-1 x 4) = 4
[4 3] = (-2 x 2) + (-1 x 3) = -7
D.n = (4, - 7)
Thus, the dot product of the two matrices D and n is determine as (4, - 7).
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03 - Describe the set S = {x : | ½x − 3| > 4} in terms of intervals.
Answer:
Hi,
Step-by-step explanation:
a)
[tex]if\ \dfrac{x}{2}-3\ > \ 0\ then\ |\dfrac{x}{2}-3|=\dfrac{x}{2}-3\\\\\dfrac{x}{2}-3 > 4\\\\\dfrac{x}{2} > 7\\\\x > 14\\[/tex]
b)
[tex]if\ \dfrac{x}{2}-3\ < \0 \ then\ |\dfrac{x}{2}-3|=-(\dfrac{x}{2}-3)=-\dfrac{x}{2}+3\\\\-\dfrac{x}{2}+3\ > \ 4\\\\-\dfrac{x}{2}\ > \ 4-3\\\\-\dfrac{x}{2}\ > \ 1\\\\\dfrac{x}{2}\ < \ -1\\\\x\ < \ -2\\[/tex]
[tex]S=\{x\in\mathbb{R}\ :\ |\dfrac{x}{2}-3|\ > \ 4\}=(-\infty,-2[\ \cup\ ]14,\infty)\\[/tex]
Describe a sequence of transformations that take isosceles trapezoid T to its image T. You may use the transformations reflection, rotation, and translation.
EARN FREE EASY POINTS BY GIVING ME THE RIGHT ANSWER!!
The types of rotation are:
Reflection: it is where the graph is inverted and each point is the same distance from the point of reflectionRotation: it is the act of rotating something around the axisTranslation: it is where every point shift by the same unitIn this case:
⇒ it is reflection since the graph is inverted and each point is the same distance from the line of reflection
Answer: reflection
Hope that helps!
There are 10000 people living in a certain city. Suppose that the rate of population growth in the city is proportional to the number of inhabitants. Suppose that 10% of the original amount increase in 30 years, how much will the population in the city after 60 years?
Answer:
.............30000.............
which choices are in the solution set of the equation below check all that apply 4x=30
The choice in the solution set of the equation 4x = 30 is x = 7.5
How to determine the solution set?The equation is given as:
4x = 30
Divide both sides of the equation by 4
4x/4 = 30/4
Evaluate the quotients
x = 7.5
Hence, the choice in the solution set of the equation 4x = 30 is x = 7.5
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Calculate the percent change if a city that used to produce 2050 kilowatt hours has reduced their consumption to 1125 kilowatt hours
The percent change if a city that used to produce 2050 kilowatt hours has reduced their consumption to 1125 kilowatt hours is 45.122% decrease.
What is percentage change?Percent increase and percent decrease are measures of percent change, which is the extent to which a variable gains or loses intensity, magnitude, extent, or value.
we know,
% change = (final- initial) / initial * 100
=(1125-2050)/1125 *100
=45.122%
Hence, percentage change be 45.122% decrease.
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This list shows the age at which 43 U.S. Presidents began their terms.
57
61
50
54
54
54
56
54
61
54
48
49
42
51
61
57
68
65
50
51
60
52
57
51
52
47
56
62
69
58
49
56
55
55
43
64
57
64
46
55
51
55
46
Before you can make a frequency table, you need to decide how to set up the age intervals. What is the BEST way to group these ages into six intervals?
a.
Age of Start Term. 20 to 30, 30 to 40, 40 to 50, 50 to 60, 60 to 70, 70 to 80.
c.
Age of Start Term. 40 to 44, 45 to 49, 50 to 54, 55 to 59, 60 to 64, 65 to 69.
b.
Age of Start Term. 20 to 29, 30 to 39, 40 to 49, 50 to 59, 60 to 69, 70 to 79.
d.
Age of Start Term. 40 to 45, 45 to 50, 50 to 55, 55 to 60, 60 to 65, 65 to 70.
Please select the best answer from the choices provided
A
B
C
D
The best intervals would be 41-45, 46-50, 51-55, 56-60, 61-65, and 69-70.
The best way to decide the appropriate interval would be to find the range of the numbers.
We have given a data set,
What is the smallest value in the data?The minimum is the smallest value in the data set. The maximum is the largest value in the data set.
What is the largest minus the smallest?This would be 69-42=27.
If you need six intervals you would need to round 27 up to 30 to be able to divide it out evenly and include all of the data.
The best intervals would be 41-45, 46-50, 51-55, 56-60, 61-65, and 69-70.
This way you are one over and one under the highest and lowest values.
Therefore, The best intervals would be 41-45, 46-50, 51-55, 56-60, 61-65, and 69-70.
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1
Select the correct answer.
Can triangle A be mapped onto triangle B using rigid transformations?
B
A
Answer: A
Step-by-step explanation:
Since the triangles are congruent by HL, we know that there is a sequence of rigid motions.
By inspection, we know this sequence involves rotating triangle A 90 degrees clockwise and then translating it right.
Find the area of the following Triangle.
Answer:
6.25cm²
Step-by-step explanation:
Hello!
The formula for the area of a triangle is [tex]A = \frac12 b*h[/tex]
A = areab = baseh = heightSolve[tex]A = \frac12 b*h[/tex][tex]A = \frac12(5)*2.5[/tex][tex]A = 2.5 * 2.5[/tex][tex]A = 6.25[/tex]The area of the triangle is 6.25 cm²
( PLEASE HELP URGENT ) A triangular prism has a height of 10 cm and the triangle has a base of 4 cm and a height of 6 cm. Find the volume of the triangular prism.
The volume of the triangular base prism is 120 centimetre cube.
How to find the volume of a triangular prism?volume of a triangular prism = BH
where
B = base areaH = height of the prismTherefore,
base area = 1 / 2 bh
base area = 1 / 2 × 4 × 6 = 24 / 2 = 12 cm
Hence,
volume of a triangular prism = 12 × 10
volume of a triangular prism = 120cm²
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Gabrielle needs to rent a car while on vacation. The rental company charges $19.95, plus 15 cents for each mile driven. If Gabrielle only has $40 to spend on the car rental, what is the maximum number of miles she can drive?
Round your answer down to the nearest mile.
Answer:
133 miles
Step-by-step explanation:
The limited budget gives rise to an inequality that can be solved for the maximum number of miles.
__
setupThe rental cost for m miles will be the sum of the fixed charges and the product of the mileage charge and the number of miles.
cost = 19.95 +0.15m
We want this to be no greater than 40, so we have the inequality ...
40 ≥ 19.95 +0.15m
solutionThis two-step inequality can be solved in the usual way:
20.05 ≥ 0.15m . . . . . step 1, subtract 19.95 from both sides
133.667 ≥ m . . . . . . . step 2, divide by the coefficient of the variable
The maximum whole number of miles Gabrielle can drive is 133.
WHAT IS THE VOLUME HELPPP!!!!!
[tex]\qquad\qquad\huge\underline{{\sf Answer}}[/tex]
Let's find the volume of the given figure ~
[tex]\qquad \sf \dashrightarrow \:Volume_{(cuboid)}= length \cdot width \cdot height [/tex]
[tex]\qquad \sf \dashrightarrow \:Volume_{(cuboid)}= 4 \cdot x \cdot x [/tex]
[tex]\qquad \sf \dashrightarrow \:Volume_{(cuboid)}= 4 {x}^{2} [/tex]
So, the correct choice is ~ C
Volume
LBH4(x)(x)4x²Option C
6x³ + 3x
how do you factor out the GCF in this question??
Hi student, let me help you out! :)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We are asked to factor out the GCF in the expression
[tex]\star~\star\mathrm{6x^3+3x}~\star~\star[/tex]
[tex]\triangle~\fbox{\bf{KEY:}}[/tex]
Divide every single term of the expression by the GCF (Greatest Common Factor).So let's divide.
But first. we should work out the GCF.
Both terms have 3x in common, so we divide both terms by 3x:
[tex]\mathrm{6x^3+3x}[/tex]
[tex]\mathrm{3x(2x^2+1)}[/tex]
To ensure that we've factored correctly, we can always use the distributive property, distribute 3x and see whether or not we end up with [tex]\mathtt{6x^3+3x}[/tex].
So the answer is
[tex]\mathrm{3x(2x^2+1)}[/tex]
Notice I put parentheses around 2x²+1. This indicates that 3x is multiplied times both 2x² and 1.
Hope it helps you out! :D
Ask in comments if any queries arise.
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~Just a smiley person helping fellow students :)
if1(2) at the level 20= 0.94770 and 1(2) at the level 21= 096020 find the required level of mortality for the estimated value of 1(2)* =0.95656.
Answer:
Step-by-step explanation
Sorry but your answer is a fake question which I can't answer
please answer asap!!!
A timer is started and a few moments later a swimmer dives into the water and then comes back up. The swimmer's depth (in feet) as
a function of time (in seconds after the timer was started) is given by the equation h(x) = ² - 14t+40.
4
Rewrite the formula in factored form and select each true statement below.
The swimmer dives to a depth of 14 feet.
The swimmer dives into the water 4 seconds after the timer was started.
The swimmer dives to a depth of 40 feet.
The swimmer comes back up 10 seconds after the timer was started.
The swimmer comes back up 40 seconds after the timer was started.
The true statement about the equations include:
The swimmer dives to a depth of 14 feet.The swimmer comes back up 10 seconds after the timer was started.How to solve the equation?The equation given is h(t) = t² - 14t + 40. We'll factorize to get the value of t. This will be:
= t² - 14t + 40
= t² - 10t - 4t + 40
= t(t - 10) - 4(t - 10)
(t - 10) = 0.
t = 0 + 10
t = 10
In this case, the swimmer dives to a depth of 14 feet and then comes back up 10 seconds after the timer was started.
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Answer:
The swimmer comes back up 10 seconds after the timer was started.
The swimmer dives into the water 4 seconds after the timer was started.
Step-by-step explanation:
The guy above me ruined my test score, thanks.
Best of luck to all who find this
These are the correct answers
The diameter of a semicircle is 6 kilometres. What is the semicircle's radius?
Answer:
3km
Step-by-step explanation:
d = 2r so r = 1/2 * d = 1/2 * 6 = 3km
If a population consisted of 75,000 people surveying 75,000 people from the population could result in a parameter but not a point estimate. True or false?
( NEED HELP ASAP! do not comment links to the answer, Would greatly be appreciative )
For number 1 I already answered it but can you tell me if I got it correct? or not ? and if I did
Not get it right, can u correct me? Thank you :)
Based on the number of children and tricycle wheels, the ratio of children to tricycle wheels is 10 : 18
Solving ratio problemsNumber of children = 10Number of tricycles = 6Number of wheels per tricycle = 3Total wheels of tricycle = Number of tricycles × Number of wheels per tricycle
= 6 × 3
= 18 wheels
Ratio of children to tricycle wheels = 10 : 18
= 5 : 9
Therefore, the ratio of children to tricycle wheels is 10 : 18
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