Answer: [tex]f^{-1} (x) = \sqrt{x} +1[/tex]
Step-by-step explanation:
[tex]y = x^2 - 2x + 1[/tex]
We know straight away the inverse will be a square root function. We also know that this inverse will have a restriction on the domain, (because you can only take the square root of a positive number).
So, to find the inverse, first we'll switch the x and y and solve for y:
[tex]x = y^2 - 2y +1[/tex]
[tex]x = (y-1)^2[/tex], (factor!)
±[tex]\sqrt{x} = y - 1[/tex]
So, the inverse "function" is:
[tex]f^-1(x)[/tex] = ±[tex]\sqrt{x} +1[/tex]
But theres an issue here!
If we tried graphing this, this "function" would not pass the vertical line test, so its not really a function at all!
We need to restrict the domain to only include the values that are above the x axis.
So our final inverse function is:
[tex]f^{-1} (x) = \sqrt{x} +1[/tex]
Expand and simplify (x + 2)(x + 1)
Answer:
x^2 + 3x + 2
Step-by-step explanation:
(x + 2) * (x + 1) =
= x * x + x * 1 + 2 * x + 2 * 1 =
= x^2 + x + 2x + 2 =
= x^2 + 3x + 2
Joan’s finishing time for the Bolder Boulder 10K race was 1.66 standard deviations faster than the women’s average for her age group. There were 410 women who ran in her age group. Assuming a normal distribution, how many women ran faster than Joan? (Round down your answer to the nearest whole number.)
Joan completed the race 1.66 standard deviations faster than the typical female of her age group. The standard deviation will be denoted by the symbol “” and the women's average finish time by the symbol “”. Then, Joan's remaining time was plus 1.66.
What is the assuming of normal distribution?The proportion of women who finished the race sooner than Joan. In this case, it is possible to apply the normal distribution.
The percentage of values to the left of z = 1.66 can then be calculated or found using a standard normal distribution table. The number of women who finished the race ahead of Joan is represented by the percentage in the table below.
According to a normal distribution table or calculator, there are approximately 0.9525 numbers to the left of the value z = 1.66.
Therefore, This means that out of the 410 female runners in Joan's age group, about 0.9525 × 410 = 391.23 of them beat her. The final count is 391 women, rounded to the next whole number.
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A salesperson's weekly paycheck is 25% more than a second salesperson's paycheck. The two paychecks total $1225. Find the amount of each paycheck. (Round your answers to the nearest cent.) first salesperson's paycheck $ second salesperson's paycheck
On solving the provided question we can say that the equation is p + 1.25p = 1225 => 2.25p = 1225 => p = 544.44
What is equation?A formula is a formula that connects two statements and uses the equal sign (=) to indicate equality. A formula that proves the equality of two formulas is called an equation in algebra. For example, in the expression 3x + 5 = 14, the equal sign separates the variables 3x + 5 and 14. The relationship between the two sentences on either side of the letter is represented by a mathematical formula. In many cases, only one variable can also act as a symbol. For example, 2x – 4 = 2.
the equation is
p + 1.25p = 1225
2.25p = 1225
p = 544.44
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There are 15 students in an art class. Mr Popplewell has 300 colored pencils to distribute equally among the students. How many colored pencils will each student receive?
Answer:
Step-by-step explanation:
15 into 300 is 20
300÷ 15 = 20
A pack of gray wolves was reintroduced to a forest. The table gives the size of
this population of wolves over time.
Which model,in which t represents that time in year,best fits the data in the table
The exponential regression equation that best fits the data in the table is given as follows:
C. f(t) = 11.9(1.12)^t.
How to obtain the exponential function?The standard definition of an exponential function is given as follows:
y = a(b)^x.
In which:
a is the value of y when x = 0.b is the rate of change.For this problem, we have that the parameters are estimated as follows:
a = 11.9, as when x = 0, y = 12, and the closest option to 12 is a = 11.9.b = 1.12, as the population increases over time, hence b > 1.This means that the correct option is given by option C.
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The length of a rectangle is 6 cm. less than 3 times the width. If the perimeter of the rectangle is 60 cm.,
find the length and the width.
(Hint - The perimeter of a rectangle is given by: P = 2L+2W)
The length is
The width is
cm.
Submit Question
cm.
Question Help: Post to forum
Answer: w=9cm l=21cm
Step-by-step explanation
l=3w-6
l+l+w+w=60
3w-6+3w-6+w+w=60
8w-12=60
8w=60+12=72
w=72/8
w=9
l=3w-6
l=27-6
l=21
The table shows the value of a camper [tex]t[/tex] years after it is purchased. It is an exponential decay function.
a. What is the value of the camper after 5 years? Round to the nearest dollar.
The value of the camper after 5 years is given as follows:
$15,155.2.
How to define the exponential function?An exponential function is defined as follows:
y = a(b)^x.
In which:
a is the value of y when x = 0.b is the rate of change.From the table, the rate of change is given as follows:
b = 29600/37000
b = 0.8.
Then the value of the camper after five years can be obtained applying the recursion, multiplying the value after 4 years by the rate of change of 0.8, as follows:
18944 x 0.8 = $15,155.2.
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Question 2
Answer: y =
<
-
Find the equation of the line with Slope = passing through the point (-35,10). Write your answer
in the form y = mz+ b.
I+
Submit Question
>
5
7
Write your answers as integers or as reduced fractions in the form A/B.
Question Help: Video Message instructor
An equation of a line that has a slope of -5/7 and passes through the point (-35, 10) is y = -5x/7 - 15.
How to determine an equation of this line?In Mathematics, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁)
Where:
m represents the slope.x and y are the points.At data point (-35, 10), a linear equation in standard form for this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 10 = -5/7(x - (-35))
y - 10 = -5/7(x + 35)
y - 10 = -5x/7 - 25
y = -5x/7 - 15
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Spring 2023
- Homework
<
Question 13, 2.4.29
Part 2 of 2
mexample
What is the area of Desert 1?
3,000,000 square miles
What is the area of Desert 2?
The area of a certain desert (Desert 1) is three times the area of another desert (Desert 2). If the sum of their areas is 4,000,000
square miles, find the area of each desert.
Get more help.
Amber McLain
HW Score: 47.92%, 11.5 of 24
points
O Points: 0 of 1
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Skill builder
Check answer
Desert 2 will be 4,000,000 square miles and Desert 1 will be 12,000,000 square miles.
How to calculate the dimensionsThe area of a shape simply means the total space that is taken by the shape. It simply expresses the extent of the region on a particular plane as well as a curved surface.
Desert 1 = 3x
Desert 2 = x.
Total area = 16,000,000
x + 3x = 16000000
4x = 16,000,000
Divide
x = 16000000 / 4
x = 40000000
Desert 2 = 4,000,000 square miles
Desert 1 = 12,000,000 square miles.
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Solve for x Each figure is a trapezoid
The value of x in the trapezoid is 12
How to determine the valueIt is important to note that the properties of a trapezoid are;
Opposite sides of a trapezoid are equalOpposite angles of a trapezoid are equalThe diagonals bisect each other at right angleThe sum of the interior angles is 360 degreesNow, let's equate the angles, we getl
7x + 21 = 105
collect the like terms, we get;
7x = 105 - 21
subtract the like terms
7x = 84
Make 'x' the subject of formula
x = 84/7
Divide the values
x = 12
Hence, the value is 12
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Solve for x with the given measures
The measured value x = 13.
What is parallelogram?
A unique variety of quadrilateral made up of parallel lines is known as a parallelogram. A parallelogram can have any angle between its adjacent sides as long as its opposite sides are parallel. If two opposite sides of a quadrilateral are parallel and congruent, it will be a parallelogram. Consequently, a quadrilateral in which both pairs of opposite sides are parallel and equal is known as a parallelogram.
We can solve this easily using alternative interior angles. A parallelogram's interior is always 360 degrees, right?
In the given parallelogram the side vy = yx
VY = YX = 80 = 6x+2
6x = 80-2=78
x = 78/6= 13
Hence The measured value x = 13.
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PLS HELP <33
Which of the numbers listed below are solutions to the equation? Check all
that apply.
|x = -25
A. 25
B. 0
C. 5
D. -5
E.-25
F. None of these
Answer:
There is no value of x that makes the equation be true since an absolute value can never be negative.
No solution
Step-by-step explanation:
How do I find the correct answer to this question?
The direction and slope of line A'B' would still be the same as that of line AB, but the positioning of A'B' would depend on the location of line AB with respect to the origin and the value of g.
What is a scale factor ?
A scale factor is a number that scales or multiplies the size or dimensions of an object or a geometric figure by the same amount in all directions. It can be used to describe changes in size or magnification of an object or figure, and is often expressed as a ratio or a percentage.
When line AB is dilated with a scale factor of 3 and a center of dilation at the origin, the resulting line A'B' will be three times as long as line AB and will pass through the origin.
The direction and slope of line A'B' will be the same as that of line AB, but it will be positioned closer to the origin.
If line AB were dilated with a scale factor of g, then the resulting line A'B' would be g times as long as line AB and would also pass through the origin.
Therefore, The direction and slope of line A'B' would still be the same as that of line AB, but the positioning of A'B' would depend on the location of line AB with respect to the origin and the value of g.
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2. Calculate the surface area and volume of a square pyramid
10cm
12cm
10cm
Answer:
Surface are = 340 cm²; Volume = 363.33 cm³.------------------------------
Full surface area is the sum of areas of 4 triangles and the square base:
S = 4*1/2*10*12 + 10² S = 240 + 100S = 340 cm²Find the height h of the pyramid, using slant height and the side of the base and Pythagorean:
h² = 12² - (10/2)²h² = 144 - 25h² = 119h = √119 ≈ 10.9 cmFind the volume:
V = 1/3*a²hV = 1/3*10²*10.9V = 363.33 cm³Consider the expansion of (3x² - k/x)^9, where k > 0.
The coefficient of the term in x^6 is 6048. Find the value of k.
(Show your work)
Answer:
The expansion of (3x² - k/x)^9 can be found using the binomial theorem. The binomial theorem states that for any real numbers a and b and any positive integer n, the expansion of (a + b)^n is given by:
(a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^(n-1)b^1 + C(n, 2)a^(n-2)b^2 + ... + C(n, n-1)a^1b^(n-1) + C(n, n)a^0b^n
where C(n, k) = n! / (k! (n-k)!).
Using this formula, we can find the coefficient of the term in x^6 in the expansion of (3x² - k/x)^9. The term in x^6 will be of the form C(9, 3) (3x²)^3 (-k/x)^6. We can find the value of C(9, 3) as follows:
C(9, 3) = 9! / (3! (9 - 3)!) = 84.
So, the coefficient of the term in x^6 is 84 * (3^3) * (-k/x)^6 = 84 * 27 * (-k)^6.
Since we are told that this coefficient is 6048, we can set these two expressions equal to each other:
6048 = 84 * 27 * (-k)^6
Dividing both sides by 84 * 27, we get:
k^6 = -6048 / (84 * 27) = -1.
Taking the sixth root of both sides, we find:
k = -1^(1/6).
So the value of k is approximately -1.122462048309372981433.
Mike is working on solving the exponential equation 37^x= 12; however, he is not quite sure where to start. Using complete sentences, describe to Mike how to solve this equation
Hint Use the change of base formula: logby=
log y over log b
Answer:
should" (and any subsequent words) was ignored because we limit queries to 32 words.
PLEASE HELP (IMAGE ATTACHED)
The answer I get is -3x^9 +2x^8 -8 over x^4
What does it simplify to? Use positive exponents only
Answer:
Step-by-step explanation:
To simplify the expression, we need to divide -3x^9 + 2x^8 - 8 by x^4.
The procedure is as follows:
Divide -3x^9 by x^4: -3x^9 / x^4 = -3x^5
Divide 2x^8 by x^4: 2x^8 / x^4 = 2x^4
Divide -8 by x^4: -8 / x^4 = -8x^-4
Combine the results from steps 1-3: -3x^5 + 2x^4 - 8x^-4
Note that the exponent of x in each term is either positive or zero. We can't simplify the expression further since we can't simplify the exponents any more.
A car is driving at 180 miles per hour. What is the speed in miles per minute
suppose that you have an alphabet of 26 letters. (a) how many possible simple substitution ciphers are there? (b) a letter in the alphabet is said to be fixed if the encryption of the letter is the letter itself. how many simple substitution ciphers are there that leave: (i) no letters fixed? (ii) at least one letter fixed? (iii) exactly one letter fixed? (iv) at least two letters fixed?
A. There are 25^26 possible simple substitution ciphers.
How did we arrive at this assertion?(a) A simple substitution cipher is a substitution of one letter for another. In a simple substitution cipher, each letter in the alphabet can be mapped to one of the 25 other letters. Therefore, for each letter in the alphabet, there are 25 possible substitutions. The total number of possible simple substitution ciphers is the number of possible substitutions for each letter, raised to the power of the number of letters in the alphabet:
25^26 = 25 × 25 × 25 × ... × 25 (26 times)
So, there are 25^26 possible simple substitution ciphers.
(b) (i) If no letters are fixed, then each letter can be mapped to one of the 25 other letters. Therefore, the number of simple substitution ciphers that leave no letters fixed is:
25 × 24 × 23 × ... × 2 × 1 = 25!
(ii) If at least one letter is fixed, then for each letter in the alphabet, there are 24 possible substitutions. Therefore, the number of simple substitution ciphers that leave at least one letter fixed is:
25 × 24 × 23 × ... × 2 × 1 = 25!
(iii) If exactly one letter is fixed, then there are 26 possible letters that could be fixed. For each of these 26 possibilities, there are 24 possible substitutions for each of the remaining 25 letters. Therefore, the number of simple substitution ciphers that leave exactly one letter fixed is:
26 × 24^25 = 26 × 24 × 24 × ... × 24 (25 times) = 26 × 24!
(iv) If at least two letters are fixed, then the number of ways to choose the two fixed letters, multiplied by the number of ways to arrange the other 24 letters, gives the number of ciphers that have exactly two fixed letters. The number of ways to arrange 24 letters is 24!, and there are 26 choose 2 ways to choose two letters from 26:
26 choose 2 × 24! = 325 × 24!
However, this counts ciphers with exactly two fixed letters twice: once for each of the two fixed letters. So, to find the number of ciphers with at least two fixed letters, we need to subtract the number of ciphers with exactly two fixed letters and divide by 2:
(25! - 325 × 24!) / 2 = (25! - 325 × 24!) / 2.
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An international company has 21,800 employees in one country. If this represents 32.2% of the company's employees, how many employees does it have in
total?
Round your answer to the nearest whole number.
employees
X
Español
?
D
Answer:
Step-by-step explanation:
The international company has 21,800 employees in one country, which represents 32.2% of the company's total employees. We can use this information to find the total number of employees by solving for the missing value.
Let's call the total number of employees in the company "X".
If the 21,800 employees in one country represent 32.2% of the total employees, we can write an equation to represent this relationship:
0.322 * X = 21,800
Next, we'll isolate X by dividing both sides of the equation by 0.322:
X = 21,800 / 0.322
Using a calculator, we can find that:
X = 68,062.5
Rounding to the nearest whole number, the company has 68,063 employees in total.
So, that's how you can use a percentage to find the total number of employees in the company. By knowing the number of employees in one country and the percentage that this represents of the total, we can find the total number of employees in the company.
4. Calculate the surface area and volume of the composite figure below.
(+3 points each: +2 work; +1 correct answer)
1.5 yd
0.9 yd
2.4 yd
1.6 yd
1.3 yd
According to the information, it can be inferred that the total surface of the figure is 23.18yr². Additionally, the volume would be 7.8 yr³
How to calculate the volume and surface of the figure?To calculate the surface of the figure we must perform the following mathematical procedure.
We must find the surface of each of the faces that the figure has, we do this by multiplying base by height; in the case of triangles it is base times height divided in two.
2.4 * 1.3 = 3.121.3 * 1.6 = 2.081.5 * 1.3 = 1.952.4 * 1.6 = 3.840.9 * 1.2 = 1.08Now we must multiply these values by the number of faces of this dimension:
3.12 * 1 = 3.122.08 * 2 = 4.161.95 * 2 = 3.93.84 * 2 = 7.681.08 * 4 = 4.32Then we add all the surfaces to have the total surface
3.12 + 4.16 + 3.9 + 7.68 + 4.32 = 23.18On the other hand, to find the volume of the figure we must multiply height by length by width; in the case of the triangle it is the same formula because it has a rectangular base.
2.4 * 1.3 * 1.6 = 4.9922.4 * 1.3 * 0.9 = 2.8084,992 + 2,808 = 7.8Learn more about yards in: https://brainly.com/question/28365360
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work and solution. Identify the error and descr
problem correctly.
Incorrect Work/Solution
m = the number of messages Nigel sent
644 = 7m
-7 -7
637 =m
Nigel sent an average of 637 messages
each day last week.
Correct Work/Solution
The correct solution to the problem is m = 92 and the error made was subtracting 7 from both sides of the equation instead of dividing both sides by 7
How to solve algebra correctly?Incorrect Work/Solution:
m = the number of messages Nigel sent
644 = 7m
subtract 7 from both sides
644 - 7= 7m - 7
637 = m
Nigel sent an average of 637 messages
each day last week.
Correct solution:
644 = 7m
divide both sides by 7
m = 644/7
m = 92
Therefore, Nigel sent an average of 92 messages
each day last week.
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I need help with the picture problem below
Using the fact that the figures are similar, we will find that MN = 12.4
How to find the length of segment MN?We know that both figures are similar, then there is a scale factor that relates them.
We know the length of two correspondent sides, these are:
HI and LM
HI = 3
LM = 18.6
if k is the scale factor, then we can wite:
3*k = 18.6
k = 18.6/3 = 6.2
Now the length of MN will be 6.2 times the length of the correspondent side on the smaller figure:
MN = 6.2*2 = 12.4
That is the lenght.
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Julie plans to invest her bonus of P100, 000.00 in two banks. The first bank earns an interest of 6% per annum while the other bank earns 8% interest per annum. If she wants her investment to earn a total interest of exactly P7, 200.00 after a year, how much does she need ot invest in each bank?
The amount of money that is invested in the first and second accounts will be 40,000 and 60,000, respectively.
What is the solution to the equation?The possible value of the variables that satisfy the equation which is known as the solution of the equation.
The interest is given as,
I = PRT / 100
Let 'x' be the amount invested in the first bank. Then the amount of investment in the second bank is (100,000 - x). Then the equation is given as,
x · 0.06 · 1 + (100,000 - x) · 0.08 · 1 = 7,200
0.06x + 8,000 - 0.08x = 7,200
-0.02x = -800
x = 40,000
The amount in the second bank is given as,
⇒ 100,000 - 40,000
⇒ 60,000
The amount of money that is invested in the first and second accounts will be 40,000 and 60,000, respectively.
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5a - 26 = -10
6a + 46 = 36
If (a,b) is the solution to the system of equations shown above, what is the value of a?
Answer:
Step-by-step explanation:
To solve for the value of "a", we can isolate "a" in one of the equations and then substitute the result into the other equation. Here's how we can do that:
Starting with the first equation:
5a - 26 = -10
5a = -10 + 26
5a = 16
a = 16 / 5
a = 3.2
Now that we know the value of "a", we can substitute it into the second equation:
6a + 46 = 36
6 * 3.2 + 46 = 36
19.2 + 46 = 36
65.2 = 36
This means that the system of equations has no solution, as the values of "a" and "b" can't both satisfy both equations at the same time.
What is the average rate of change? Problem in the picture
The average rate of change over the interval (-1, 2) of the function is 3.
What is a parabola?A parabola is a curve drawn in a plane. Where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix).
A graph of the parabola is given.
The vertex is (-1, -4).
And the y-intercept is -3.
So, the equation of the parabola is,
y = (x + 1)² - 4.
The average rate of change over the interval (-1, 2):
= f(-1) - f(2)/(-1 - 2)
= (-4 - 5)/-3
= 3
Hence, 3 is the average rate of change.
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Given right triangle ABC with altitude
BD drawn to hypotenuse AC. If
AD = 12 and BD = 36, what is the
length of AC?
A
B
12D
36
Answer: AC =
Submit Answer
C
+
Can someone help and find the length of AC?
If AD = 12 and BD = 36 in the right triangle ABC with altitude BD was drawn to hypotenuse AC, then the length of AC is 12√82.
What is the similarity law for triangles?
It is defined as the law to prove that two triangles have the same shape, but it is not compulsory to have the same size. The ratio of the corresponding sides is in the same proportions and the complementary angles are congruent.
We can use the property of similar triangles to find the length of AC.
Since BD is an altitude of triangle ABC, we can write:
(AD)(DC) = (BD)²
Substituting the given values, we get:
12(DC) = 36²
Solving for DC, we get:
DC = (36)²/12 = 3 x 36 = 108
Now, using the Pythagorean theorem, we can write:
(AC)² = (AD)² + (DC)²
Substituting the given values, we get:
(AC)² = 12² + 108²
Simplifying and solving for AC, we get:
AC = √(12² + 108²) = √(12² x (1 + 9²)) = 12 x √(1 + 9²) = 12 x √(82)
Therefore, If AD = 12 and BD = 36 in right triangle ABC with altitude
BD was drawn to hypotenuse AC, then the length of AC is 12√82.
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how to find the perimeter of a right angled triangle by only its hypotenuse
Answer:
Step-by-step explanation:
To find the perimeter of a right-angled triangle when only the length of the hypotenuse is known, you'll need to use the Pythagorean theorem. Here's a step-by-step explanation:
Step 1: Understand the Pythagorean theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b):
c^2 = a^2 + b^2
Step 2: Substitute the known length of the hypotenuse
Let's say the length of the hypotenuse is "h". Substitute "h" for "c" in the Pythagorean theorem:
h^2 = a^2 + b^2
Step 3: Solve for one of the unknown sides
To find the perimeter of the triangle, we need to know the lengths of both sides "a" and "b". To do this, we need to solve for one of the unknown sides. Let's solve for "a".
Square root both sides of the equation:
√(h^2) = √(a^2 + b^2)
h = √(a^2 + b^2)
Square both sides of the equation:
h^2 = a^2 + b^2
Step 4: Use the Pythagorean theorem to find the other unknown side
Now that we have an expression for "a", we can use the Pythagorean theorem again to find "b":
h^2 = a^2 + b^2
Substitute the expression for "a" that we found in Step 3 into the equation:
h^2 = (√(a^2 + b^2))^2 + b^2
h^2 = a^2 + b^2 + b^2
h^2 = 2 * b^2 + a^2
h^2 - a^2 = 2 * b^2
b^2 = (h^2 - a^2) / 2
Take the square root of both sides:
b = √((h^2 - a^2) / 2)
Step 5: Calculate the perimeter
The perimeter of the triangle is the sum of the lengths of all three sides, so we can now calculate the perimeter using the lengths of "a" and "b" that we found in Steps 3 and 4:
P = a + b + h
Step 6: Conclusion
Therefore, the perimeter of the right-angled triangle can be found by first using the Pythagorean theorem to find the lengths of two of the sides, and then adding up the lengths of all three sides to find the perimeter.
Use Descartes' Rule of Signs to determine the possible numbers of positive and negative real zeros.
Answer:
(a) Possible number(s) of positive real zeros: 3
(b) Possible number(s) of negative real zeros: 1
Step-by-step explanation:
Descartes' Rule of Signs tells us the maximum number of positive and negative roots.
Positive root case
[tex]f(x)=-3x^4+2x^3-x^2+9x+7[/tex]
As there are 3 sign changes, the maximum possible number of positive roots is 3.
Negative root case
[tex]\begin{aligned}f(x)&=-3(-x)^4+2(-x)^3-(-x)^2+9(-x)+7\\&=-3x^4-2x^3-x^2-9x+7 \end{aligned}[/tex]
As there is one sign change, the maximum possible number of negative roots is 1.
Find the lower quartile for the data. {63, 74, 63, 76. 73, 70, 74, 69, 69, 65, 70, 70, 72, 70, 65}
A. 65
B. 68
C. 66
D. 67
Answer:
65
Step-by-step explanation:
The lower quartile for the given data is 65. To calculate this, we first need to sort the data in ascending order: {63, 63, 65, 65, 69, 70, 70, 70, 70, 72, 73, 74, 74, 76}. The lower quartile is then calculated as the median of the lower half of the data, which is 65.
