2) y = -5 +4V7-2 A) Domain: { All real numbers. } Range: { All real numbers. } B) Domain: x 22 Range: y 2-5 + C) Domain: 'x z 2 Range: ys-5 D) Domain: x 2-2 Range: y z 5
Answers
Looking at the restrictions over the variable x, we know that the domain is:
[tex]x\ge2[/tex]To find the range, notice that:
[tex]\sqrt[]{x-2}\ge0[/tex]On the other hand, the function:
[tex]y=\sqrt[]{x-2}[/tex]is an increasing function (its value grows when x grows), and can get as large as we want provided a sufficiently large value for x. Then, the range of such a function would be:
[tex]y\ge0[/tex]Which does not get altered when we multiply the square root of (x-2) by 4.
Since the function:
[tex]y=-5+4\sqrt[]{x-2}[/tex]is a 5-units shift downwards, then the variable y can take any value from -5 onwards.
Then, the range of the function is:
[tex]y\ge-5[/tex]Another way to find the range is to isolate x from the equation:
[tex]\begin{gathered} y=-5+4\sqrt[]{x-2} \\ \Rightarrow y+5=4\sqrt[]{x-2} \\ \Rightarrow\frac{y+5}{4}=\sqrt[]{x-2} \\ \Rightarrow(\frac{y+5}{4})^2=x-2 \\ \Rightarrow x-2=(\frac{y+5}{4})^2 \\ \Rightarrow x=(\frac{y+5}{4})^2+2 \end{gathered}[/tex]Since we already know that x must be greater than 2, then:
[tex]\begin{gathered} 2\le x \\ \Rightarrow2\le(\frac{y+5}{4})^2+2 \\ \Rightarrow0\le(\frac{y+5}{4})^2 \\ \Rightarrow0\le|\frac{y+5}{4}| \\ \Rightarrow0\le|y+5| \end{gathered}[/tex]From here, there are two options:
[tex]\begin{gathered} 0\le y+5 \\ \Rightarrow-5\le y \\ \text{ Or} \\ 0\le-y-5 \\ \Rightarrow y\le-5 \end{gathered}[/tex]Since we know an equation for y, then:
[tex]\begin{gathered} -5\le-5+4\sqrt[]{x-2} \\ \Rightarrow0\le4\sqrt[]{x-2} \end{gathered}[/tex]Or:
[tex]\begin{gathered} -5+4\sqrt[]{x-2}\le-5 \\ \Rightarrow4\sqrt[]{x-2}\le0 \end{gathered}[/tex]The second case is not true for every x.
Therefore:
[tex]-5\le y[/tex]Therefore:
[tex]\begin{gathered} \text{Domain: }x\ge2 \\ \text{Range: }y\ge-5 \end{gathered}[/tex]Related Questions
Your credit card charges 16.5% interest on any unpaid balance. If your credit card balance forthe month is $650, how much will you be charged in interest and what will your new balancebe?
Answers
Solve for interest
[tex]\begin{gathered} \text{convert the rate first from percentage to decimal} \\ 16.5\%\rightarrow0.165 \\ \\ \text{Then multiply it to the balance to get the interest} \\ \$650\times0.165=\$107.25 \\ \\ \text{Therefore, the interest is \$107.25} \end{gathered}[/tex]Solve for new balance.
[tex]\begin{gathered} \text{Add the interest the the credit card balance to get the new balance.} \\ \$650+\$107.25=\$757.25 \\ \\ \text{Therefore, the new balance will be \$757.25} \end{gathered}[/tex]From an elevation of 0.5 m below the surface of the water, a swimmer dives at a rate of 0.25 m/s. What is the depth of the swimmer after 4 minutes?A. -60.5 mB. 60.5 mC. -1.5 mD. 1.5 m
Answers
By assuming that the rate is constant and the speed is given by
[tex]\text{speed = }\frac{dis\tan ce}{time}[/tex]we have that
[tex]0.25\frac{m}{s}=\frac{d}{240s}[/tex]because 4 minutes is equal to 240 seconds. Then, the distance is given as
[tex]\begin{gathered} d=(0.25\frac{m}{s})(240s) \\ d=60\text{ m} \end{gathered}[/tex]Therefore, the depth of the swimmer will be 60 meters plus 0.5 meters. Then, the answer is -60.5 meters (option A)
how do I solve (-2+5i) (4-i) +(2-i)
Answers
Given:
[tex](-2+5i)(4-i)+(2-i)[/tex]Using the distributive property for the multiplication then combine the like terms
so, the given expression will be:
Note: i² = -1
[tex]\begin{gathered} (-2+5i)(4-i)+(2-i) \\ =-2(4-i)+5i(4-i)+(2-i) \\ =-8+2i+20i-5i^2+2-i \\ =-8+2i+20i+5+2-i \\ =(-8+5+2)+(2i+20i-i) \\ =-1+21i \end{gathered}[/tex]So, the answer will be:
[tex]-1+21i[/tex]Consider the right triangle shown below that has side lengths of x, y and r units.For each of the following questions, write an expression in terms of θ that answers the question. Enter "theta" for θ.y is how many times as large as r? ____times as large x is how many times as large as r? _____times as large y is how many times as large as x? ___times as large
Answers
Determine relation between side of triangle by using trigonometry.
[tex]\begin{gathered} \sin \theta=\frac{y}{r} \\ y=r\sin \theta \end{gathered}[/tex]So y is sin theta times as large as r.
[tex]\begin{gathered} \cos \theta=\frac{x}{r} \\ x=r\cos \theta \end{gathered}[/tex]So x is cos theta times as large as r.
[tex]\begin{gathered} \tan \theta=\frac{y}{x} \\ y=x\tan \theta \end{gathered}[/tex]So y is tan theta times as lasrge as x.
6. A concert promoter's profit is p(n) = 61n - 5500, where n is the number of tickets sold. Find theprofit for the promoter if he sells 850 tickets. Show your work and explain what your answer means interms of the problem. (5 points)6
Answers
Given:prfit p(n)=61n-5500
Find: profit when he sells 850 tickets.
Explanation: n=850
put in the given equation of profit , we get
[tex]\begin{gathered} p(n)=61\times850-5500 \\ =51850-5500 \\ =46350 \end{gathered}[/tex]Final answer: the profit is 46350.
Triangle JKL is graphed below. If triangle JKL is translated 4 units down and 1 unit to the left, what will be the coordinates of L’ ? A. (-3,1) B. (-1,3) C. (1,-3) D. (3,-1)
Answers
J = (-2, 4) K = (3, 4) L = (2, 1)
Translated 4 units down
J = (-2, 4 - 4) K = (3, 4 - 4) L = (2, 1 - 4)
J = (-2 , 0) K = (3, 0) L = (2, -3)
Translated 1 unit to the left
J = (-2 -1, 0) K = (3 - 1, 0) L = (2 - 1, -3)
J = (-3. 0) K = (2, 0) L = (1, -3)
Convert the unit to the specified equivalent unit round your answer to at least1 decimal place if necessary
Answers
Given:
There are given that the 209 ounces to convert into the decigram.
Explanation:
According to the concept:
To convert the ounces into the decigram, we need to multiply the mass values by 283.
That means,
The value of 1 ounce is 283 decigram
So,
[tex]\begin{gathered} 209ounces=209\times283decigram \\ =59147decigram \end{gathered}[/tex]Final answer:
Hence, the value of the 209 ounces is 59147 decigrams.
There are 221 6th graders in the HPS district that are going on field trip to the Detroit Institute of Arts when we get back to school. If each bus holds 46 students, how many buses should be hierd to transport us?
Answers
Students = 221
Each bus holds 46 students
Divide the total number of students (221) by the capacity of each bus (46 )
221 /46 = 4.8 = 5 buses
Find the absolute change and the percentage change for the given situation.120 is decreased to 18
Answers
The absolute change is defined as:
[tex]V_2-V_1[/tex]where V1 and V2 are the initial and final values, respectively.
Plugging the values given we have that:
[tex]18-120=-102[/tex]Therefore the absolute change is -102. (The minues sign indicate a decrease)
The percentage change is given by:
[tex]\frac{V_2-V_1}{\lvert V_1\rvert}\cdot100[/tex]plugging the values given we have:
[tex]\frac{18-120}{\lvert120\rvert}\cdot100=-\frac{102}{120}\cdot100=-0.85\cdot100=85[/tex]Therefore the percentage change is -85% (Once again the minus sign indicate a decrease)
use the formula for computing future value using compound interest to determine the value of an account at the end of 10 years if a principal amount of $10,000 is deposited in an account at an annual rate of 3% and the interest is compounded quarterly. The amount after ten years will be $_______ (round to the nearest cent as needed). A = P ( 1 + r/m) ^ n
Answers
Given:
P=$10000 : m=4 : r=0.03 : n=40
[tex]A=P(1+\frac{r}{m})^n[/tex][tex]A=10000(1+\frac{0.03}{4})^{40}[/tex][tex]A=10000(1.0075)^{40}[/tex][tex]A=10000(1.3483)[/tex][tex]A=\text{ \$13483}[/tex][tex]1\text{ \$= 100 cent}[/tex][tex]A=13483\times100[/tex][tex]A=\text{ 1348300 cent}[/tex]Therefore,The amount after ten years will be 1348300 cent.
Select the expressions that equivalent to 8(4r)
Answers
The expression that is equivalent to 8(4r) is 32r.
The given expression is 8(4r)
The given expression is an algebraic expression with constant and variables, here, r is the variable, and 4 and 8 are constant terms.
We need to simplify the expression as much as possible.
In order to simplify it, we need to solve the brackets, when we open the brackets, the terms get multiplied by each other,
So, here we go,
8(4r) = 8* 4r = 32r
This is the simplest form that we can have,
So, 8(4r) = 32r
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HelpClassify each number below as a rational number or as an irrational number
Answers
By definition:
- Rational numbers are those numbers that can be written as simple fractions. A fraction has this form:
[tex]\begin{gathered} \frac{a}{b} \\ \end{gathered}[/tex]Where "a" is the numerator and "b" is the denominator. Both are Integers, and:
[tex]b\ne0[/tex]- Irrational numbers cannot be written as simple fractions.
Then, knowing those definitions, you can identify that:
1. The number:
[tex]-\sqrt[]{25}=-5[/tex]Since -5 is an Integer, it can be written as:
[tex]=\frac{-5}{1}[/tex]Therefore, it is a Rational Number.
2. You can identify that the second number is a Repeating Decimal because the line over the decimal digits indicates that its digits are periodic.
By definition, Repeating Decimals are Rational Numbers.
3. Notice that the next number is:
[tex]-\sqrt[]{10}\approx-3.162278[/tex]Since it cannot be written as a simple fraction, it is not a Rational Number.
4. For the number:
[tex]-\frac{18}{5}[/tex]You can identify that it is a fraction whose numerator and denominator and Integers. Then, it is a Rational Number.
5. Notice that the last number is:
[tex]18\pi[/tex]By definition, π is an Irrational Number.
Therefore, the answer is:
If I work 4weeks on and one week off how many weeks per year
Answers
We know that the year has 52 weeks
From 5 weeks, you work 4. So, we can use the rule of three
Number of weeks Weeks of work
5 4
52 x
[tex]\begin{gathered} 5x=4\cdot52 \\ 5x=208 \\ x=\frac{208}{5} \\ x=41.6\text{ w}eeks \\ or\text{ 41 complete w}eeks\text{ } \end{gathered}[/tex]Sowen rolled two number cubes with sides numbered 1 through 6, 20 times. Her sums are recorded in the table below.49899462.1012879111087935What is the experimental probability of rolling a sum of 9?4/20Сь5/20Od4/365/36
Answers
The total number of experiment, N=20.
From the given data, 9 is obtained 5 times.
The number of times of getting 9, n=5.
Hence, the probability of getting a sum of 9 is,
[tex]\begin{gathered} P=\frac{n}{N} \\ P=\frac{5}{20} \end{gathered}[/tex]Hence, option b is correct.
Write a sequence of rigid motions to take figure CBA to figure MLK
Answers
We have the following:
• The first thing is to ,rotate, in the opposite direction of the hands of the clock, in such a way that it is the same.
,• Then make a ,translation, from left to right until it overlaps one on the other
I need help with number 16 can I please get help?
Answers
We can break apart the figure into 4 separate figures and find the area of each of these individual figures. Then sum to get area of total figure.
We break apart the figure as shown below:
First,
Area of Triangle = 0.5 * base * height
Area of Rectangle = base * height
Now, let's find each of the Areas A through D:
Area of A:
To find the base of this triangle, we have to use pythagorean theorem. By which we can write:
[tex]b^2+4^2=6^2[/tex]Where b is the base. let's solve for b:
[tex]\begin{gathered} b^2+16=36 \\ b^2=36-16 \\ b^2=20 \\ b=\sqrt[]{20} \\ b=\sqrt[]{4}\sqrt[]{5} \\ b=2\sqrt[]{5} \end{gathered}[/tex]Area is
[tex]\begin{gathered} 0.5\cdot\text{base}\cdot\text{height} \\ =\frac{1}{2}\cdot2\sqrt[]{5}\cdot4 \\ =4\sqrt[]{5} \end{gathered}[/tex]Area of B:
This is a rectangle with base = 10 and height 4, so the area is:
Area = 4 * 10 = 40
Area of C:
Area of C is exactly same as area of B, base is 10 and height is 4. So,
Area = 4 * 10 = 40
Area of D:
Like area of A, we have to find the base of the triangle first, using pythagorean theorem. We can write:
[tex]\begin{gathered} b^2+4^2=4.5^2 \\ \end{gathered}[/tex]Solving for b:
[tex]\begin{gathered} b^2+4^2=4.5^2 \\ b^2+16=20.25 \\ b^2=\frac{17}{4} \\ b=\frac{\sqrt[]{17}}{\sqrt[]{4}} \\ b=\frac{\sqrt[]{17}}{2} \end{gathered}[/tex]Now, area of triangle is:
[tex]A=\frac{1}{2}(\frac{\sqrt[]{17}}{2})(4)=\sqrt[]{17}[/tex]Area of whole figure:
[tex]\begin{gathered} 4\sqrt[]{5}+40+40+\sqrt[]{17} \\ =80+4\sqrt[]{5}+\sqrt[]{17} \end{gathered}[/tex]Since mulch costs $3 per square feet, we have to multiply the area by "3", so we have:
[tex]3\times(80+4\sqrt[]{5}+\sqrt[]{17})\approx279.202[/tex]It will cost around:
$279.20
the letter "t" estimated makes up 10% of a language. a random sample of 700 letters is taken from a book. what is the approximate probability that the random sample of 700 letters will contain 8.9% t's
Answers
In this problem, we have a Binomial Probability Distribution
so
n=700
x=700*(8.9/100)=62.3=62
[tex]P(x=62)=\frac{700!}{62!(700-62)!}\cdot0.10^{(62)}\cdot0.90^{(700-62)}[/tex]P(x=62)=0.0313
the aprroximate probability is 3.13%L) Point A bisects CR. CA = 8x + 1 and AR = 6x+13.Find CA.34927
Answers
We will investigate some application of line bisectors.
We have a line segment denoted as ( CR ). A point ( A ) is said to be bisecting the line segment.
Line bisection involves the process of equally dividing the entire line segment in two equal halves! We can also represent this scenario graphically for clearification:
We can also represent line bisection mathematically in terms of two smaller divisions ( CA and AR ) as follows:
[tex]Bi\sec tion\colon\text{ |CA| = |AR| = }\frac{|CR|}{2}[/tex]Where, the quantities are expressed as magnitudes/lengths of each segment indicated.
We are given expressions for bifurcated line segments |CA| and |AR| in terms of variable ( x) as follows:
[tex]\begin{gathered} |CA|\text{ = 8x + 1} \\ |AR|\text{ = 6x + 13} \end{gathered}[/tex]Now we will use the expression given for each smaller division of line |CR| and plug it in the general " Bisection " expression developed above:
[tex]\begin{gathered} |CA|\text{ = |AR|} \\ 8x\text{ + 1 = 6x + 13} \end{gathered}[/tex]We have constructed an equation with a single variable ( x ). We can solve this equation for the variable ( x ) using basic mathematical operations as follows:
[tex]\begin{gathered} 8x\text{ - 6x = 13 - 1} \\ 2x\text{ = 12} \\ x\text{ = 6} \end{gathered}[/tex]Once we have solved for the variable ( x ). We will again use the defined expression for each smaller segments and determine the magnitudes as follows:
[tex]\begin{gathered} |CA|\text{ = 8}\cdot(6)\text{ + 1 } \\ |CA|\text{ = 49 units} \\ \\ |AR|\text{ = 6}\cdot(6)\text{ + 13 } \\ |AR|\text{ = 49 units} \\ \\ |\text{ CR | = 2}\cdot|CA|\text{ = 2}\cdot49 \\ |\text{ CR | = 98 units} \end{gathered}[/tex]Roberto bought a new graduated cylinder for his chemistry class. it holds 650 mililiters of liquid. if the cylinder has a radius of 5 cm, then how tall is the cylinder.
Answers
8.27 cm
Explanation
Step 1
the volume of a cylinder is given by:
[tex]\begin{gathered} \text{Volume}=\text{ area of the circle}\cdot\text{ height} \\ \text{Volume}=(\pi\cdot raidus^2)\cdot\text{height} \end{gathered}[/tex]then
Let
Heigth=unknown=h
volume=650 ml
radius= 5 cm
also
1 mililiter= 1 cubic centimeter
so,replacing
[tex]\begin{gathered} \text{Volume}=(\pi\cdot raidus^2)\cdot\text{height} \\ \text{650 }=(\pi\cdot(5cm)^2)\cdot\text{h} \\ 650=25\pi\cdot h \\ \text{divide both sides by 25 }\pi \\ \frac{650}{25\pi}=\frac{25\pi h}{25\pi} \\ h=\frac{650}{25\pi}=8.27\text{ cm} \end{gathered}[/tex]so, the cylinder is 8.27 cm tall.
I hope this helps you
Convert percent to fraction and simplify if possible 11% =
Answers
To convert percent to fraction divide it by 100
Then to change 11% to a fraction divide 11 by 100
[tex]\frac{11}{100}[/tex]Since 11 and 100 can not divide by the same number, then
11% = 11/100 in the simplest form
The mean life of a television set is 97 months with the variance of 169. If a sample of 59 televisions is randomly selected what is the probability that the sample mean would be less than 100.9 months? Round your answer to four decimal places if necessary
Answers
Given that the mean life of a television set is 97 months, you can set up that:
[tex]\mu=97[/tex]You also know that the variance is:
[tex]\sigma^2=169[/tex]You can find the standard deviation by taking the square root of the variance. Then:
[tex]\sigma=\sqrt{169}=13[/tex]You need to find:
[tex]P(X<100.9)[/tex]You need to find the z-score with this formula:
[tex]z=\frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]Knowing that:
[tex]\bar{X}=100.9[/tex]You can substitute values into the formula and evaluate:
[tex]z=\frac{100.9-97}{\frac{13}{\sqrt{59}}}\approx2.30[/tex]You have to find:
[tex]P(z<2.30)[/tex]Using the Standard Normal Distribution Table, you get:
[tex]P(z<2.30)\approx0.9893[/tex]Then:
[tex]P(X<100.9)\approx0.9893[/tex]Hence, the answer is:
[tex]P(X<100.9)\approx0.9893[/tex]weir the equation of a line for the following problems
Answers
The equation of line in slope intercept form is given by:
[tex]y=mx+c[/tex]Where m is the slope and c is the y-intercept
It is given that the slope of the line is 0 and the y-intercept is -3 so it follows:
[tex]m=0,c=-3[/tex]Hence the equation is given by:
[tex]\begin{gathered} y=0x-3 \\ y=-3 \end{gathered}[/tex]Hence y=-3 is the equation of the line ith
902 degrees Celsius round the temperature to the nearest ten
Answers
The temperature is 92 degrees celsius. Rounding 92 degrees celsius to the nearest ten will be 90 degrees celsius. The closest ten to 92 is definitely 90.
Therefore,
[tex]92^{\circ}\approx90^{\circ}(To\text{ the nearest tens)}[/tex]Which car traveled at the slowest rate? a.750 miles in 10 hours b. 800 miles in 12 hours c. 450 miles in 6 hours d.1000 miles in 14 hours e. 1000 miles in 14 hours
Answers
The car with the smallest speed (or slowest rate) exists the car with 800 miles in 12 hours.
What is meant by rate of speed?The ratio of the distance traveled and the time it took to travel that distance yields the rate at which cars move (also known as the speed).
To determine each unit rate (number of miles traveled in 1 hour). To determine the unit rate, we will divide the distance by the time.
rate = distance/time
Substitute the values in the above equation, we get
Then, for each car we will the speeds:
a) S = 750 mi/10 h = 75 mi/h
b) S = 800 mi/12 h = 66.66 mi/h
c) S = 450 mi/6 h = 75 mi/h
d) S = 1000 mi/14 h = 71.428 mi/h
Then we can see that the car with the smallest speed (or slowest rate) is the car with 800 miles in 12 hours.
Therefore, the correct answer is option b) 800 miles in 12 hours.
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if Sarah has 98 baseball cards in her collection for birthday her she's given 78 new cards and she sold 108 cards how many baseball cards does Sarah have left
Answers
68
1) Gathering the data
Sarah has 98 cards
She's given 78 then we can add
98 +78 = 176 cards
2) And right after that Sarah sold 108 cards, so now we can write
176 -108 = 68
3) Finally, Sarah now has 68 baseball cards left.
Elena and Diego each wrote equations to represent these diagrasolve it. You can assume that angles that look like right anglesa35°to
Answers
∠35 and ∠w are vertical angles, therefore:
[tex]\angle w=35[/tex]Since ∠w and ∠x are supplementary, we can conclude:
[tex]\begin{gathered} \angle w+\angle x=180 \\ 35+x=180 \end{gathered}[/tex]Therefore:
Diego has written the correct equation.
im using goformative to answer my question and I already did number 7 but the rest show my answer is incorrect so I will appreciate if I can get a help from you, I will paste the image of the work.
Answers
To get side QR we can use the the tangent function:
[tex]\begin{gathered} \tan 37=\frac{9}{QR} \\ QR=\frac{9}{\tan37} \\ QR=11.9 \end{gathered}[/tex]Now, to find PR we can use the sine function:
[tex]\begin{gathered} \sin 37=\frac{9}{PR} \\ PR=\frac{9}{\sin 37} \\ PR=15 \end{gathered}[/tex]Finally to find the remaining angle we have to remember that the interior angles of any triangle have to add to 180, then angle P is 53°.
10^4 divided by 10^6 in expanded form, then find out what the answer is with a single power.
Answers
The given expression is:
[tex]\frac{10^4}{10^6}[/tex]Exapanding the above expression,
[tex]\frac{10\times10\times10\times10}{10\times10\times10\times10\times10\times10}[/tex](The number of 10s are the same as the power of 10).
Now, cancel out the common terms in the numerator and the denominator of the above expression.
[tex]\frac{1}{10\times10}[/tex]The denominator of the above expression can now be expressed as the power of 10 as,
[tex]\frac{1}{10^2}[/tex]According to the law of exponents,
[tex]\frac{1}{x_{^{^m}}}=x^{-m}[/tex]Hence, we can write
[tex]\frac{1}{10^2}=10^{-2}[/tex]Therefore, 10^4 divided by 10^6 can be expressed as a term with a single power as,
[tex]10^{-2}[/tex]Amount of money in Ivanna's account and the amount of money in Brian's account and write the equation
Answers
Given
Starting amount of Ivanna = $180
put per month = $70
Starting amount of Brian = 0
put per month = $100
Find
Amount of money in Ivanna's account
the amount of money in Brian's account
equation to show when the two accounts have the same amount of money
Explanation
Let x be the number of months after today.
so, according to the question ,
Amount of money in Ivanna's account = starting amount + amount per month.
so , 180 + 70x
amount of money in Brian's account = 0 + 100x = 100x
equation to show when the two accounts have same amount of money =
[tex]180+70x=110x[/tex]so ,
[tex]\begin{gathered} 180=110x-70x \\ 180=40x \\ x=4.5 \end{gathered}[/tex]so , in 4.5 months both have same money
Final Answer
Hence ,
Amount of money in Ivanna's account = 180 + 70x
Amount of money in Brian's account = 110x
Equation to show when the two accounts have same amount of money =
[tex]180+70x=110x[/tex]I need help find the answer to number 10 and 11
Answers
10. We need to identify the value of b in the function:
[tex]f(x)=b^x[/tex]so that it produces the graph given.
Observing the graph, we see that the point (1,2) lies on it. Thus, we have:
[tex]\begin{gathered} 2=b^1 \\ \\ \Rightarrow2=b \end{gathered}[/tex]Therefore, the value of the base b is: 2