Determine the inverse of the function by interchanging the variables and solving for y in terms of X
Answers
We are required to find the inverse of the function
The first step is to interchange the variable x for y
[tex]x=\frac{y}{2}-\frac{3}{2}[/tex]The next step is to make y the subject of the formula
[tex]\begin{gathered} x=\frac{y}{2}-\frac{3}{2} \\ \frac{y}{2}=x+\frac{3}{2} \\ \text{ Multiply the equation throughout by 2} \\ y\text{ = 2x + 3} \end{gathered}[/tex]The answer is y = 2x + 3
Related Questions
Moshde runs a hairstyling business from her house. She charges $42 for a haircut and style. Her monthly expenses are $1070. She wants to be able to put at least $1,249 per month into her savings account order to open her own salon. How many "cut & styles" must she do to save at least $1,249 per month?
Answers
ANSWER:
56 cut & styles
STEP-BY-STEP EXPLANATION:
They tell us that each haircut and style charges $42 and that the monthly expenses are $1070, he wants to save a total of $1249, with this information we can establish the following equation:
[tex]42x-1070=1249[/tex]Where x would be the amount of cut & styles, we solve for x:
[tex]\begin{gathered} 42x=1249+1070 \\ x=\frac{2319}{42} \\ x=55.2 \\ x\cong56 \end{gathered}[/tex]If you make a total of 55 cut & styles, the amount does not reach a total of $1249 per month, therefore, at least 56 cut & styles are needed, to achieve the monthly goal.
Given the vectors u =-7j and w=-9i+4j, find 8u and u+w.Write your answers in the form ai+bj.
Answers
Recall that:
[tex]\begin{gathered} \text{For all a, b, c, d, e real numbers:} \\ (ai+bj)+(ci+dj)=(a+c)i+(b+d)j, \\ e(ai+bj)=(ea)i+(eb)j\text{.} \end{gathered}[/tex]Therefore:
[tex]\begin{gathered} 8u=8(-7j)=8(0i-7j)=(8\cdot0)i+(8\cdot(-7))j=0i-56j=-56j\text{.} \\ u+w=(-7j)+(-9i+4j)=(0i-7j)+(-9i+4j)=(0-9)i+(-7+4)j \\ =-9i-3j\text{.} \end{gathered}[/tex]Answer:
[tex]\begin{gathered} 8u=-56j\text{.} \\ u+w=-9i-3j\text{.} \end{gathered}[/tex]Find the cardinal number of the setWhere N denotes the set of all natural numbers
Answers
And x is divisible by 6 . The natural number of x will be
[tex]x=\mleft\lbrace36,42,48,54\mright\rbrace[/tex]3. A toy box is 24 cm long, 15 cm wide and 11 cm high. What is the volume of the toy box? What is the correct number sentence for this problem? A.V=24×15×11B.V=24×15C.V=24×11D.V=15×11
Answers
ANSWER
[tex]\begin{gathered} V=24*15*11 \\ V=3960\text{ }cm^3 \end{gathered}[/tex]EXPLANATION
The box is a rectangular prism. The volume of a rectangular prism is given by:
[tex]V=L*W*H[/tex]where L = length
W = width
H = height
Therefore, the volume of the box can be written in the number sentence:
[tex]V=24*15*11[/tex]and the volume of the box is:
[tex]V=3960\text{ }cm^3[/tex]That is the answer.
Hi, what is the LCM of the numbers 3 and 15
Answers
Answer:
15
Step-by-step Explanation:
LCM is the least common multiple, that is, is the least number that is a multiple of both numbers.
To find it, first, let's write the multiples of 3:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Now, let's write the multiples of 15:
Multiples of 15: 15, 30, 45, ...
If you compare the multiples of 3 and 5, we can see that they have some common multiples, as 15 and 30.
From this common multiples, 15 is the smallest number. So, 15 is the LCM of 3 and 15.
Answer: 15.
slove equations with variables on both sides-4k - 10 = -5k
Answers
We will investigate how to solve an equation consisting of one variable
We have the following equation at hand:
[tex]-4k\text{ -10 = -5k}[/tex]The basic rule applied in solving equation like above is mathematical operations. We apply basic operations like:
[tex]\text{adding, subtracting, multiplying, division}[/tex]on both sides of the equation accompained by a variable or a number in an attempt to isolate the variable ( k ).
To isolate the variable ( k ) we need all the terms involving the variable ( k ) on one side of the equation.
We will add ( 4k ) on both sides of the equation as follows:
[tex]\begin{gathered} -4k\text{ -10 + 4k= -5k + 4k} \\ (\text{ 4k - 4k ) - 10 = -k} \\ -10\text{ = -k} \end{gathered}[/tex]Now to remove the negative sign accompained by ( k ) on the right hand side of the equation. We wil multiply both sides with ( -1 ) as follows:
[tex]\begin{gathered} -1\cdot(-10)\text{ = -1}\cdot(-k) \\ 10\text{ = k} \end{gathered}[/tex]Hence, the value of ( k ) is:
[tex]10[/tex]
(4.7 x 10-3) x 351Simplify the expressionusing scientific notation and express your answer(2.5 x 10') < (3.3 X 100)in scientific notation. Round your answer to the nearest thousandth.AnswerKeypadKeyboard Shortcutsx10
Answers
Given:
[tex]\frac{(4.7\times10^{-3})\times351}{(2.5\times10^5)\times(3.3\times10^6)}[/tex]Remove the brackets and multiply common terms
[tex]\begin{gathered} \frac{(4.7\times10^{-3})\times351}{(2.5\times10^5)\times(3.3\times10^6)} \\ =\frac{4.7\times10^{-3}\times351}{2.5\times10^5\times3.3\times10^6} \\ =\frac{4.7\times351\times10^{-3}}{2.5\times3.3\times10^5\times10^6} \\ =\frac{1649.7\times10^{-3}}{8.25\times10^{11}} \end{gathered}[/tex]Simplify further to get
[tex]\begin{gathered} \frac{1649.7\times10^{-3}}{8.25\times10^{11}} \\ =\frac{16497\times10^{-1}_{}\times10^{-3}}{825\times10^{-2}\times10^{11}} \\ =\frac{16497\times10^{-4}}{825\times10^9} \end{gathered}[/tex]This further gives
[tex]\begin{gathered} \frac{16497\times10^{-4}}{825\times10^9} \\ =\frac{16497}{825}\times\frac{10^{-4}}{10^9} \\ =19.996\times10^{-4-9} \\ =19.996\times10^{-14} \end{gathered}[/tex]Therefore, the answer is
[tex]19.996\times10^{-14}[/tex]Find the height of the cliff. If necessary, round to the nearest hundredth yard.
Answers
We are given a diagram showing a slope, and a vertical height. We now have what represents a right angled triangle. The distance from the base of the cliff to the end of the slope is given as 24 yards. The slope itself is 37 yards. We shall now determine the height of the cliff (from ground to top) as indicated.
Note that we shall use the Pythagoras' theorem which is;
[tex]c^2=a^2+b^2[/tex]Where we have
[tex]\begin{gathered} c=\text{hypotenuse (longest side)} \\ a,b=\text{other sides} \end{gathered}[/tex]We can now substitute the given values/side lengths and we'll have;
[tex]37^2=24^2+b^2[/tex][tex]1369=576+b^2[/tex]Subtract 576 from both sides;
[tex]793=b^2[/tex]Take the square root of both sides;
[tex]\begin{gathered} \sqrt[]{793}=\sqrt[]{b^2} \\ 28.160255\ldots=b \end{gathered}[/tex]Rounded to the nearest hundredth, the answer now becomes;
ANSWER:
[tex]b=28.16yd[/tex]The last option is the correct answer
Type the correct answer in the box. Use numericals instead of words. 5 less than a number is equivalent to 1 more than three times the number. The number is _____.
Answers
Answer:
2
Explanation:
Let the number be x
5 less than a number is expressed as x - 5
1 more than three times the number is expressed as 3x + 1
Equate both expression and find the number
x - 5 = 3x+1
x - 3x = 1 - 5
-2x = -4
x = -4/-2
x = 2
Hence the number is 2
Write an expression for the operation described.
"5 divided by the product of 3 and 2"
A (5 ÷ 3) × 2(5 ÷ 3) × 2
B 3 × (2 ÷ 5)3 × (2 ÷ 5)
C (3 × 2) ÷ 5(3 × 2) ÷ 5
D 5 ÷ (3 × 2)
Answers
D] 5 ÷ (3 × 2) is the expression for the operation "5 divided by the product of 3 and 2".
The operation "5 divided by the product of 3 and 2" means that number 5 divided by the product of 3 and 2.
The mathematical representation of this operation is 5 ÷ (3 × 2).
The answer to this operation = 5 / 6.
Hence, 5 ÷ (3 × 2) is the expression for the operation "5 divided by the product of 3 and 2".
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What is the solution to the equation below? Round your answer to two decimal places.ln x = 0.2A.x = 1.58B.x = -0.70C.x = -1.61D.x = 1.22
Answers
Given the equation:
[tex]\ln \left(x\right)=0.2[/tex]Apply the properties of logarithms:
[tex]e^{ln(x)}=e^{0.2}[/tex]Simplify:
[tex]x=e^{0.2}=1.22[/tex]Answer: D. x = 1.22
h(x) =x² +9 if h(x)=9 , x =
Answers
The given expression as; h(x) =x² +9
for h(x) = 9
Substitute the value of h(x) = 9 in the given expression;
h(x) =x² +9
9 =x² +9
x² = 9 - 9
x² = 0
x = 0
Answer : x = 0
in the diagram, ab to ec are perpendicular. if m
Answers
Given A = {(1, 3X-1, 5}(6, 4)), B = {(2, 0X4, EX-4, 5x0, 0)) and C = {(1, 1x0, 2x0, 3)(0, 4X-3, 5)), answer the following multiple
choice question:
From the list of sets A, B, and C above, choose the set of relations that correctly represents a function.
O Set A only
O Sets A and C only
O Sets A and B only
Answers
The functions is Set A and Set B.
What is meant by function?A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.
A relation in a set is said to be a function, if every first element of an ordered pair in a set is related with a unique element of a second element.
No, two distinct second elements of an ordered pair, has the same first element.
For, example, {(1,2), (1,3), (4,5)}, is not a function, but it is a relation.
In Ordered pair, (x, y)
x=First Element
y= Second Element
→In Set A
First Element Second Element
1 3
-1 5
6 4
Every First element of set A has a unique second element. So, it is a function.
→In Set B
First Element Second Element
2 0
4 6
-4 5
0 0
Every First element of set B has unique second element and no two distinct Second element of set B, has same first element. So, it is a function.
→In Set C
First Element Second Element
1 1
0 2
0 3
-3 5
As, two same first elements of set C has distinct second element. So, it is not a function.
Therefore, Set A and Set B, are functions .
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Amelia bought spider rings for Halloween goodie bags. She bought 13 packs of red rings, 16 packs of yellow rings, and 14 packs of green rings. If each pack had 12 rings, how many rings did Amelia buy?
Answers
We know that
• She bought 13 packs of red rings.
,• She bought 16 packs of yellow rings.
,• She bought 14 packs of green rings.
,• Each pack has 12 rings.
This problem is about multiplication, notice that each pack includes 12 rings, that means we need to multiply each pack by 12, in order to find the total number of rings of each color.
[tex]R=13\cdot12=156[/tex]There are 156 red rings.
[tex]Y=16\cdot12=192[/tex]There are 192 yellow rings.
[tex]G=14\cdot12=168[/tex]There are 168 green rings.
Now, we sum all these numbers to find the total
[tex]T=168+192+156=516[/tex]Therefore, there are 516 rings in total.A 51-inch TV suggests that the main diagonal of the TV is 51 inches. Determine the dimensions of the screen of a 51 -inch TV with a 16:9 aspect ratio.Please see attached photo
Answers
The aspect ratio 16:9 indicates the next relation between x and y:
[tex]\frac{y}{x}=\frac{16}{9}[/tex]Applying the Pythagorean theorem to the right triangle formed:
[tex]51^2=x^2+y^2[/tex]Isolating y from the first equation:
[tex]y=\frac{16}{9}x[/tex]Substituting in the second equation:
[tex]\begin{gathered} 51^2=x^2+(\frac{16}{9}x)^2 \\ 2601=x^2+(\frac{16}{9})^2x^2 \\ 2601=x^2+\frac{16^2}{9^2}^{}x^2 \\ 2601=x^2+\frac{256}{81}^{}x^2 \\ 2601=\frac{337}{81}^{}x^2 \\ 2601\cdot\frac{81}{337}=x^2 \\ 625.166172=x^2 \\ \sqrt[]{625.17}\approx x \\ 25\approx x \end{gathered}[/tex]Replacing in the equation of y:
[tex]\begin{gathered} y=\frac{16}{9}\cdot25 \\ y\approx44.44 \end{gathered}[/tex]The approximate dimensions are:
length = 25 in
height = 44.44 in
1/9=_/54What is the answer?
Answers
the line contains the point (-3,5) and is perpendicular to the line y=3x-4
Answers
two lines are perpendicular when the multiplication of their slopes is equal to -1. The slope of y = 3x - 4 is 3. Then the slope of a perpendicular line is:
[tex]\begin{gathered} m\cdot3=-1 \\ m=-\frac{1}{3} \end{gathered}[/tex]Slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. Replacing with point (-3, 5) and m = -1/3, we get:
5 = -1/3(-3) + b
5 = 1+ b
5 - 1 = b
4 = b
Then, the equation is:
y = -1/3x + 4
the relationship between the minutes a candle is burned and the size of the candle in millimeters is shown on the graph.
Answers
The function is a decreasin line so the more time goes the side will decrease so the correct answer is:
The candle started at 9mm and shrinks 5mm every 4 minutes
Differentiate a trig function that is greater than a power of 1, and involve either quotient, chain, or product rule.Differentiate a sine and cosine function that involves product and chain rule. Find the equation of the tangent line at x = a special triangle point (i.e. /4, /6, /3).Differentiate a function that involves both trig and exponential functions.[hint: add your own twist to this question for level 3/4]Differentiate an exponential function. [hint: add your own twist to this question for level 3/4]Differentiate a function where you have “y” and “x” on both sides of the equation and they cannot be simplified by collecting like terms or isolating y (i.e. y on one side and y^2 on the other). [hint: add your own twist to this question for level 3/4]
Answers
Solution:
Given a trigonometric function that is greater than power of 1 as shown below:
[tex]y=sin^2x\text{ ---- equation 1}[/tex]To differentiate the function, we use the chain rule.
According to the chain rule,
[tex]\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}[/tex]From equation 1, let
[tex]u=sin\text{ x --- equation 2}[/tex]This implies that
[tex]\begin{gathered} y=u^2 \\ \Rightarrow\frac{dy}{du}=2u \end{gathered}[/tex]From equation 2,
[tex]\begin{gathered} \begin{equation*} u=sin\text{ x} \end{equation*} \\ \Rightarrow\frac{du}{dx}=cos\text{ x} \end{gathered}[/tex][tex]\begin{gathered} Recall\text{ from the chain rule:} \\ \frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx} \\ \Rightarrow2u\times\text{cos x} \\ \frac{dy}{dx}=2ucos\text{ x} \\ but\text{ } \\ u=sin\text{ x} \\ \therefore\frac{dy}{dx}=2(sin\text{ }x)(cos\text{ }x) \end{gathered}[/tex]Prison Sentences The average prison sentence for a person convicted of second-degree murder is 15 years. If the sentences are normally distributed with astandard deviation of 2.1 years, find the following probabilities.P (x> 18) =
Answers
Givens.
• The mean is 15 years.
,• The standard deviation is 2.1 years.
,• x = 18.
Using a graphic calculator, the probability P(X > 18) is 0.0766.
Therefore, the answer is 0.07
Annie's backyard deck cost $61.75 per square meter to build. The deck is 7 meters wide and14 meters long. How much did it cost to build the deck?
Answers
ANSWER
the cost to build the deck is $6051.5
EXPLANATION
Given that;
The length of the deck is 14 m
The width of the deck is 7m
1 m^2 is equivalent to $61.75
Follow the steps below to find the cost to build the deck
Step 1; Find the area of the deck
[tex]\begin{gathered} \text{ Recall, that the deck is a rectangular shape} \\ \text{ Area of a rectangle = length }\times\text{ width} \\ \text{ Area of a reactangle = 14 }\times\text{ 7} \\ \text{ Area of a rectangle = 98m}^2 \end{gathered}[/tex]Step 2; Find the total cost of the deck
Let x represents the total cost to build the deck
[tex]\begin{gathered} \text{ 1m}^2\text{ }\rightarrow\text{ \$61.75} \\ \text{ 98m}^2\text{ }\rightarrow\text{ \$x} \\ \text{ cross multiply} \\ \text{ 1m}^2\text{ }\times\text{ \$x = \$61.75 }\times\text{ 98m}^2 \\ \text{ Isolate \$x }\frac{}{} \\ \text{ \$x = }\frac{\text{ \$61.75}\times98\cancel{m^2}}{1\cancel{m^2}} \\ \text{ \$x = \$61.75 }\times\text{ 98} \\ \text{ \$x = \$6051.5} \end{gathered}[/tex]Therefore, the cost to build the deck is $6051.5
find the volume round to the nearest tenth use 3.14 for pi 5km
Answers
Step 1
List all parameters
[tex]\begin{gathered} \pi\text{ = 3.14} \\ r\text{ = 5km} \\ \end{gathered}[/tex]Step 2
Write the volume of a sphere
[tex]undefined[/tex]Thomas is married and files jointly with his spouse. Their combined taxable income is $25,799. Their employers withheld $4,386 in taxesfor the year. Determine theamount to be refundedor the balance due.Circle one: RefundBalance Due
Answers
EXPLANATION
As we can see on the table, the amount to be refunded is equivalent to the difference between $3,866 and $4,386, so it is $520
Translate to a system Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than $500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is $3 and for a package the cost is $7.
Answers
Given:
Let x be the number of the cards and y be the number of the package.
Given that the number of cards is at least 4 more than twice the number of packages.
[tex]x=2y+4[/tex]Given that mailing costs no more than $500 and the cost of mailing a card is $3 and for a package, the cost is $7.
[tex]3x+7y=500[/tex]Substitute x=2y+4 in this equation, we get
[tex]3(2y+4)+7y=500[/tex][tex]6y+8+7y=500[/tex][tex]13y=500-8[/tex][tex]y=\frac{492}{13}[/tex][tex]y=37.8[/tex]Let y=37 and substitute in x=2y+4, we get
[tex]x=2\times37+4[/tex][tex]x=78[/tex]Hence the number of cards = 78 and the number of packages =37.
The total cost for this is $493 not more than $500.
An online company is advertising a mixer on sale for 35% off the original price of $224.99 what is the sale price for the mixer? Round your answer to the nearest cent, if necessary.
Answers
Given:
The original price of mixer is $224.99.
The discount on the mixer is 35%.
Explanation:
Determine the discount amount on the mixer.
[tex]\begin{gathered} d=\frac{35}{100}\cdot224.99 \\ =78.7465 \end{gathered}[/tex]Determine the sale price of the mixer.
[tex]\begin{gathered} 224.99-78.7465=146.2435 \\ \approx146.24 \end{gathered}[/tex]So sale price of the mixer is $146.24.
Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval 2 ≤ x ≤ 6.
Answers
To find the average rate of change over an interval we need to calculate how much the function has changed over that interval by subtracting the final value by the initial one and dividing by the lenght of the interval. With this in mind we have:
[tex]\begin{gathered} \text{rate}=\frac{19-13}{6-2} \\ \text{rate}=\frac{6}{4} \\ \text{rate}=1.5 \end{gathered}[/tex]The average rate of change for this interval is 1.5
Vanessa collected Barbie dolls. She began with 2 dolls and added the same amount of dolls to her collection each year. In the 24th year, Vanessa had 98 dolls. Which function, d(n), can be used to determine the number of dolls Vanessa had in any year?
Answers
The correct answer is d(n) = 4n +2
Simplify.4n + 12 + 7n4 n + 1923 n16 n +711 n+ 12
Answers
11 n+ 12
In this expression, to simplify means to reduce it to the simpler expression. Hence:
1) Grouping similar terms
4n + 12 + 7n =
2) Adding them up:
4n+7n+12=
11n +12
Using the hottest and coolest months data, find the equation for line of best fit for this data showing all steps by hand.
Answers
Let
x -----> average temperature
y ----> Electricity Bill
we take the points
(99,150) and (69,80)
step 1
Find out the slope
m=(80-150)/(69-99)
m=-70/-30
m=7/3
step 2
Find out the equation of the line in slope-intercept form
y=mx+b
we have
m=7/3
point (69,80)
substitute and solve for b
80=(7/3)(69)+b
b=80-161
b=-81
the equation is
y=(7/3)x-81
using a graphing tool
Remember that the value of y cannot be a negative number