Which function has the following characteristics? • A vertical asymptote at x = 3 • A horizontal asymptote at y = 2 Domain: {** +3] 2x - 8 X - 3 y=x2-9 2 9 x² - 4 4 OB. V C. 2x2 - 18 x² - 4 4 2x2 - 8 O D. ** - 9
Answers
SOLUTION
To get this, note that the vertical asymptote can be gotten by setting the denominator to be equal to 0.
If we do this, we will notice that the vertical asymptote of option A and option D is x = 3
That is
for option A
[tex]\begin{gathered} y=\frac{2x-8}{x-3} \\ x-3=0 \\ x=3 \end{gathered}[/tex]For option D
[tex]\begin{gathered} y=\frac{2x^2-8}{x^2-9} \\ x^2-9=0 \\ x^2=9 \\ x=3 \end{gathered}[/tex]So, the answer is either option A or D.
But to get the correct answer, let us look at the graphs for both functions
Graph of A
[tex]y=\frac{2x-8}{x-3}[/tex]From the graph, you can see that the domain is defined at x = 3. Notice that the green line cut across x = 3.
Now let's check Graph of option D
[tex]y=\frac{2x^2-8}{x^2-9}[/tex]From the graph, you can see that the domain is defined at x = -3 and x = 3. Notice that the purple and green line cut across x = -3 and x = 3. So, the domain here is
[tex]x=\pm3[/tex]Hence
Related Questions
Zacaria's kayaks store annual rent and fixed cost are $ 15,600. The variable cost each kayak old is $315 and the price of each kayak is $425.e. What is the number of kayaks that must be sold to obtain $ 50,000 in the profits?
Answers
The profit function can be gotten using the formula below:
[tex]P(x)=R(x)-C(x)[/tex]where R(x) is the revenue function and C(x) is the cost function.
As previously calculated, the revenue and cost functions are given to be:
[tex]\begin{gathered} R(x)=425x \\ C(x)=315x+15600 \end{gathered}[/tex]Therefore, the profit function is:
[tex]\begin{gathered} P(x)=425x-(315x+15600) \\ P(x)=425x-315x-15600 \\ P(x)=110x-15600 \end{gathered}[/tex]The profit to be made is $50,000. Equating the profit function to this amount, we can get the number of kayaks required. This is shown below:
[tex]\begin{gathered} 50000=110x-15600 \\ 110x=50000+15600 \\ 110x=65600 \\ x=\frac{65600}{110} \\ x=596.4 \end{gathered}[/tex]Therefore, the number of kayaks that must be sold is approximately 597 kayaks to make $50,000 in profits.
Find 5x3 + 2y2 - 9, where x = 4 and y = 2
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The equation can be solved when we substitute the value of x and y accordingly .
[tex]\begin{gathered} 5(4)^3+2(2)^2\text{ - 9} \\ 5\text{ }\times\text{ 64 + 2 }\times\text{ 4 - 9} \\ 320\text{ + 8 - 9} \\ =319 \end{gathered}[/tex]An airplane pilot over the Pacific sights an atoll at the angle of depressionof 5º. At this time, the horizontal distance from the airplane to the atoll is4,535 meters. What is the height of the plane to the nearest meter?
Answers
Answer:
397 meters
Explanation:
Let us draw a diagram to understand the situation better.
Where H is the height of the plane.
Now the tan ratio for 5° gives
[tex]\tan 5^o=\frac{opposite}{\text{adjacent}}[/tex]Since opposite = H and adjacent = 4535 meters, the above gives
[tex]\tan (5^o)=\frac{H}{4535}[/tex]Multiplying both sides by 4535 gives
[tex]\tan (5^o)\times4535=H[/tex]Since tan(5°) =0.0875.., ( just used a calculator to find out) the above becomes
[tex]0.0875\times4535=H[/tex][tex]H=396.8125[/tex]rounding to the nearest meter gives
[tex]\boxed{H=397.}[/tex]2x -10 < 6 solve the inequality[tex]2x - 10 \leqslant 6[/tex]
Answers
The given equation is:
[tex]2x\text{ - 10 }\leq\text{ 6}[/tex]To simply this, just keep the x term on the left side and others at the right side. Therefore,
[tex]2x\text{ }\leq\text{ 6 + 10}[/tex][tex]2x\text{ }\leq\text{ 16}[/tex][tex]x\text{ }\leq\text{ }\frac{16}{2}[/tex]or
[tex]x\text{ }\leq\text{ 8}[/tex]Hence, the solution is x =< 8.
What is an equation of the line that passes through the point (1,-3) and is perpendicular to the line x+3y=21
The answer I need begins with y =
Please do not answer if your solution doesn't end that way, thank you.
Answers
The linear equation that is perpendicular to the line x+3y=21 is:
y = 3*x - 6
How to find the equation of the line?A general line in the slope-intercept form is written as:
y = m*x + b
Where m is the slope and b is the y-intercept.
Two linear equations are perpendicular if the product between the two slopes is equal to -1.
Rewriting the given line we can get:
x +3y = 21
3y = 21 - x
y = 21/3 - x/3
y = (-1/3)*x + 21/3
Then the slope is (-1/3), if our line is perpendicular to this one, then:
m*(-1/3) = -1
m = 3
our line is:
y = 3*x + b
To find the value of b, we use the fact that our line passes through (1, - 3)
-3 = 3*1 + b
-3 - 3 = b
-6 = b
The line is y = 3*x - 6
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Four people buy raffle tickets and place all of their tickets into a hat. Raphael bought 8 tickets, Leonardo bought 2 tickets, Michelangelo bought 5 tickets, and Donatello bought 11 tickets. If a person randomly selects one raffle ticket from the hat, what is the probability that Raphael purchased the ticket? Write your answer as a decimal and round three decimal places.
Answers
0.308
Explanation
The probability of an event is the number of favorable outcomes divided by the total number of outcomes possible,so
[tex]P(A)=\frac{favorable\text{ outcomes }}{total\text{ outcomes}}[/tex]so
Step 1
favourable outcome is the result that is desired, so as the result is a that Raphael purchase the ticket
a) Let
[tex]favorable\text{ outcome=}8[/tex]and , the total outcome is the total ticket,so
[tex]\begin{gathered} total\text{ outcome=Raphael+Leonardo+Michelangelo+Donatello} \\ total\text{ outcome=8+2+5+11=26} \end{gathered}[/tex]b) now, replace in the formula:
[tex]\begin{gathered} P(A)=\frac{favorable\text{ outcomes }}{total\text{ outcomes}} \\ P(A)=\frac{8}{26} \\ P(A)=0.3076923 \\ rounded \\ P(A)=0.308 \end{gathered}[/tex]therefore,the answer is 0.308
I hope this helps you
Write a system of equations that could be used to determine the number of commercials and the number of movies on which Alexa songs were played Define the variables that you use to write the system.
Answers
We make a system of equations from the statement
Alexa will earn $20 every time one of her songs is played in a commercial and she will $110 every time one of her songs is played in a movie.
[tex]T=20c+110m[/tex]
where T is the total earned, c the number of commercials and m the number of movies
Alexa earned a total of $340 in 8 commercials and movies
we replace the total (340) on the first equation and make and equation for the total of commercials and movies
[tex]\begin{gathered} 340=20c+110m \\ c+m=8 \end{gathered}[/tex]it is our system
Solve the inequality AND graph the solution set. 7(4y - 8) < -5(y-2)
Answers
A solution to the given inequality is y < 2, which is represented by the graph shown in the image attached below.
What is a graph?In Mathematics, a graph simply refers a type of chart that is typically used for the graphical representation of data on both the horizontal and vertical lines of a cartesian coordinate or a number line.
Next, we would solve the given inequality by simplifying it as follows:
7(4y - 8) < -5(y - 2)
Opening the bracket, we have the following:
28y - 56 < -5y + 10
Rearranging the inequality by collecting like terms, we have the following:
28y + 5y < 10 + 56
33y < 66
Dividing both sides of the inequality by 33, we have the following:
y < 66/33
y < 2
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Perform (3x3 – 2x2 + 3x – 4) ÷ (x – 3) to find the value of the remainder.Question 3 options:A) 98B) 88C) 68D) 78
Answers
The remainder of the division can be found below
The answer is definitely C.
which equation , together with y=-1.5x+3 , makes a system with just on solution.y = -1.5x+6y = -1.5x2y = -3x+62y + 3x = 6y = -2x + 3
Answers
EXPLANATION
Given the equation y= -1.5x + 3
In order to get a system with just one solution, the viable options are those that allow both lines to intersect in just one point.
Those equations are:
y= -1.5x
y=-2x + 3
The system of equations y=-1.5x+3 and y=-1.5x+6 has no solution because subtracting both expressions give us the result 3=0
The system y=-1.5x+3 and 2y = -3x+6, or in the same way the system y=-1.5x+3 and 2y + 3x = 6 has no solution because they have the same expression.
Answer the following. (a) Find an angle between 0 and 2π that is coterminal with -13T/12 (b) Find an angle between 0° and 360° that is coterminal with 810°. Give exact values for your answers.
Answers
(a) Find an angle between 0 and 2π that is coterminal with -13π/12.
To get coterminal angles, we simply have to add or subtract 2π. In this problem, we are looking for a coterminal angle that is between 0 and 2π, so we will add 2π to -13π/12:
[tex]\frac{-13\pi}{12}+2\pi=\frac{11\pi}{12}[/tex](b) Find an angle between 0° and 360° that is coterminal with 810°.
Similarly to the previous exercise, we will substract 360° to 810° to find a coterminal angle that is between 0 and 360°:
[tex]\begin{gathered} 810\text{\degree}-360\text{\degree}=450\text{\degree} \\ 450\text{\degree}-360\text{\degree = 90\degree} \end{gathered}[/tex]ANSWER
a. 11π/12
b. 90°
what is 19,593 in expanded form?
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The extended form of the given number is:
[tex]10000+9000+500+90+3[/tex]Sketch the graphs of each of the following functions showing all steps on the same set of axe
Answers
Part D
we have the function
[tex]\begin{gathered} y=\frac{4}{-(x+1)}+3=\frac{4-3(x+1)}{-(x+1)}=\frac{4-3x-3}{-(x+1)}=\frac{1-3x}{-(x+1)}=\frac{3x-1}{x+1} \\ \\ y=\frac{3x-1}{x+1} \end{gathered}[/tex]In this rational function
Remember that
The denominator cannot be equal to zero
so
The value of x cannot be equal to x=-1
At x=-1 there is a vertical asymptote
Find out a horizontal asymptote
Degree on Top is Equal to the Bottom
so
the horizontal asymptote is at y=3/1=3
Find out the intercepts
y-intercept (value of y when the value of x=0)
For x=0
[tex]y=\frac{3(0)-1}{0+1}=-1[/tex]The y-intercept is (0,-1)
Find out the x-intercept (value of x when the value of y=0)
For y=0
[tex]\begin{gathered} 0=\frac{3x-1}{x+1} \\ \\ 3x-1=0 \\ 3x=1 \\ x=\frac{1}{3} \end{gathered}[/tex]The x-intercept is (0.33,0)
With the given information
Graph the function
using a graphing tool
see the figure below
Bella works at a popular clothing store. Last winter, her manager asked her to track thestore's sweater sales. This box plot shows the results.Price of sweaters sold ($)304050607080What fraction of the sweaters cost $50 or less?
Answers
From the given graph, it appears that the price of the sweaters sold ranges in 3 bars - from $45 up $60.
Sweaters costing $50 or less covers 1 out of the 3 bars in the graph. In its equivalent fraction, we get:
[tex]undefined[/tex]The bill for the meal was $24.00. Dario left a tip of 16 2/3%. How much was the tip?
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We have to calculate the tip.
It is the 16 2/3% of $24.
We start by converting 16 2/3% to decimals:
[tex](16+\frac{2}{3})\%\cdot\frac{1}{100\%}=\frac{16.6667}{100}=0.166667[/tex]Now, we can calculate the tip by multiplicating this proportion by 24:
[tex]\text{tip}=0.166667\cdot24=4[/tex]Answer: the tip is $4.
There are 1300 people over the age over the age of 12 in the small town. If the same rate of cell phone usage continues , how many yrs will it takes for 1300 people have a cell phone.
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We have to find how many years it will take for the 1300 people to have a cellphone.
This means we have to find x for f(x) = 1300, that is "the number of people using cellphones at time x is 1300".
Then, we can write:
[tex]\begin{gathered} f(x)=1300 \\ 880(1.03)^x=1300 \\ 1.03^x=\frac{1300}{880} \\ 1.03^x\approx1.477 \\ x\cdot\ln (1.03)=\ln (1.477) \\ x=\frac{\ln (1.477)}{\ln (1.03)} \\ x\approx\frac{0.39}{0.03} \\ x\approx13.2 \end{gathered}[/tex]Answer: it will take 13.2 years for the 1300 people to have a cellphone.
6. 5.6% of 283 is what number?7. 55% sign of 104 is what number?
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To solve the exercise you can use a rule of three:
[tex]\begin{gathered} 283\rightarrow100\text{\%} \\ x\rightarrow5.6\text{\%} \end{gathered}[/tex][tex]\begin{gathered} x=\frac{5.6\text{\%}\cdot283}{100\text{\%}} \\ x=\frac{5.6\cdot283}{100} \\ x=\frac{1584.8}{100} \\ x=15.848 \end{gathered}[/tex]Therefore, 5.6% of 283 is 15.848.
Under her cell phone plan, Ella pays a flat cost of $58.50 per month and $3 per gigabyte, or part of a gigabyte. (For example, if she used 2.3 gigabytes, she would have to pay for 3 whole gigabytes.) She wants to keep her bill under $65 per month. What is the maximum whole number of gigabytes of data she can use while staying within her budget?
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She can consume up to 2 gigabytes of data at a time without exceeding her monthly budget.
Given that Ella pays a one-time fee of $58.50 and has to pay $3 per gigabyte used. She must pay $3xg if she utilizes g gigabytes.
We now need to determine how many gigabytes of data she can utilize in total while staying within her budget.
She is spending less than $65.
For this, we can construct an inequality statement as shown below.
58.50+3g< 65
3g<65-58.50
3g<6.5
g<6.5/3
g<2.163.
Therefore, she should only use a total of 2 gigabytes to keep the cost within her budget.
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6 - 2x = 5x - 9x + 20
Answers
Answer:
x = 7
Step-by-step explanation:
6 - 2x = 5x - 9x + 20
−2x + 6 = −4x + 20
2x + 6 = 20
2x = 14
x = 14 : 2
x = 7
(2x³-7x² + 7x-2) ÷ (x-2)
Answers
Answer:
Please look at the attachment below, thanks! :D
Step-by-step explanation:
The system of equations(4x-y = 4(x+1) hasly = 6The system of equations 4x-y=4(x+1) , y = 6 has:A. one solutionB. infinitely many solutionsC. no solution
Answers
EXPLANATION
Plugging in y=6 into the first equation:
[tex]4x-6=4(x+1)[/tex]Applying the distributive property:
[tex]4x-6=4x\text{ +4}[/tex]Subtracting -4x to both sides:
[tex]4x-4x-6=4[/tex]Adding +6 to both sides:
[tex]4x-4x=4+6[/tex]Adding like terms:
[tex]0\ne10[/tex]In conclusion, the system of equations has no solutions.
A store is having a 20% off sale on all merchandise if my buys one item and see if $13 what was the original price of a purchase
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Since all merchandise has 20% off, so the original price is multiplied by 0.8 (80%).
Let's use the variable x to represent the original price. So we can write the following equation and solve it for x:
[tex]\begin{gathered} 0.8\cdot x=13 \\ x=\frac{13}{0.8} \\ x=16.25 \end{gathered}[/tex]Therefore the original price is $16.25.
I need help I'm still having a bit of trouble with multiplying radicals
Answers
The area of the rectangle is;
[tex]36\sqrt[]{3\text{ }}\text{ sq cm}[/tex]Here, we want to find the area of the rectangle
To do this, we will have to multiply the length of the sides
Mathematically, we have this as follows;
[tex]\begin{gathered} \text{Area = Length }\times\text{ Width} \\ \text{Area = 6 cm }\times\text{ 6}\sqrt[]{3\text{ }}\text{ cm} \\ =\text{ 36}\sqrt[]{3}\text{ sq cm} \end{gathered}[/tex][tex]\begin{gathered} 6\times6\text{ = 36} \\ \sqrt[]{6}\text{ }\times\text{ 6 = 6}\sqrt[]{6} \\ \sqrt[]{6}\text{ }\times\text{ }\sqrt[]{6}\text{ = 6} \\ 6\sqrt[]{6}\text{ }\times\text{ 6}\sqrt[]{6}\text{ = (6}\times6)\times(\sqrt[]{6\text{ }}\text{ }\times\text{ }\sqrt[]{6}\text{ ) = 36}\times6\text{ = 216} \\ \sqrt[]{3}\text{ }\times\text{ }\sqrt[]{2\text{ }}\text{ = }\sqrt[]{\text{ (3}\times2)}\text{ = }\sqrt[]{6} \\ 7\sqrt[]{3}\text{ }\times\text{ 8}\sqrt[]{7\text{ }}\text{ = (7}\times8\text{ )}\times\text{ (}\sqrt[]{3}\text{ }\times\sqrt[]{7\text{ }}\text{ ) = 56}\sqrt[]{21} \end{gathered}[/tex]1/2(x+6) = 2x and 5-2x =7x-h
Answers
Expression given:
[tex]\frac{1}{2}(x+6)=2x[/tex][tex]5-2\cdot(\frac{4}{5})=7\cdot(\frac{4}{5})-h[/tex]Procedure:
0. Solving for ,x: ,multiplying both sides by 2
[tex]2\cdot\frac{1}{2}(x+6)=2x\cdot2[/tex][tex]x+6=4x[/tex]Substracting both sides by x
what is the slope - intercept form of a line that passes through points(2,12) and (4,17) ?
Answers
A line pass through points (2, 11) and (4, 17);
The slope-intercept form of a line is given as;
[tex]\begin{gathered} y=mx+c \\ \text{Where m = slope} \\ c=\text{ y-intercept} \\ \\ \text{But slope m=}\frac{y_2-y_1}{x_2-x_1} \\ \text{Where x}_1=2 \\ x_2=4 \\ y_1=11 \\ y_2=17 \\ m=\frac{17-11}{4-2}=\frac{6}{2} \\ m=3 \end{gathered}[/tex]Also, for the y-intercept c;
[tex]\begin{gathered} \text{But when x=2, y=11} \\ \text{let's use this point to get the c} \\ 11=3(2)+c \\ c=11-6 \\ c=5 \\ \end{gathered}[/tex]Thus, the slope intercept equation of the line is;
y = 3x + 5
8. An elephant weighs 74,416 ounces. How many tons does it weigh?
Answers
To know how many tons does the elephant weigh, we have to do a convertion:
1 ounce=3.125x10^-5 ton
[tex]74.416ounces\times\frac{3.125\times10^{-5}}{1\text{ ounce}}=2.3255\times10^{-3}[/tex]The elephant weighs 2.3255x10^-3 tons
If anyone to solve this problem it will be greatly appreciated and I will be sure to give the best possible rating
Answers
Step 1
Graph the points from the table and make a plot that can help determine the possible line of best fit.
Hence, the equation of the line of best fit with values to the nearest hundredths will be;
[tex]y=0.06x+2.57[/tex]Express the confidence interval 0.25 < p < 0.43 in the form of p ± E.
Answers
The confidence interval written as 0.25 < p < 0.43 in the form p ± E is; CI = 0.34 ± 0.09
How to express confidence Interval?The general formula for confidence interval is;
CI = x' ± z(s/√n)
where;
CI is confidence interval
x' is sample mean
z is confidence level value
s is sample standard deviation
n is sample size
We are given the confidence interval as;
0.25 < p < 0.43
To write it in the form p ± E means that;
- p = (up confed limit) + (low conf limit)/2
- E= (up confed limit) - (low confed limit)/2
p = (0.25 + 0.43)/2
p = 0.34
E = (0.43 - 0.25)/2
E = 0.09
Thus;
CI = 0.34 ± 0.09
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if <5 = 130 and m<3 - 13x. find the value of x Round to the nearest tenth if necessary.
Answers
From the figure, it can be seen that ∠3 and ∠5 are co-interior angles. Thus, we find that the value of x is 3.84.
It is given to us that -
∠5 = 130
and, ∠3 = 13x
We have to find out the value of x.
Here, we have two parallel lines. Parallel lines are the lines that do not intersect at any points.
There is also a transversal that passes through the two parallel lines in the same plane intersecting the parallel lines at two separate points.
When a transversal cuts two parallel lines, there are different angles formed as shown in the figure.
∠3 and ∠5 are co-interior angles that lie on the same side of the transversal.
So, ∠3 + ∠5 = 180
=> 13x + 130 = 180
=> 13x = 50
=> x = 3.84
Thus, from the value of the sum of co-interior angles, we find that the value of x is 3.84.
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A 5000-seat theater has tickets for sale at $25 and $40. How many tickets should be sold at each price for a sellout performance to generate a total revenue of $149,000? The number of tickets for sale at $25 should be ____ The number of tickets for sale at $40 should be ____
Answers
Let us call x the number of $25 seats and y $40 seats, then we know that there are in total 5000 seats in a theatre; therefore, we have the equation
[tex]x+y=5000[/tex]Also, after selling this many tickets the total revenue should be $149, 000; therefore, we get the equation
[tex]25x+40y=149,0000[/tex]Hence, we have a system of equations with two equations and two unknowns.
We solve this system by substitution.
First, we solve for x in the first equation to get
[tex]x=5000-y[/tex]we then put this into the second equation to get
[tex]25(5000-y)+40y=149,000[/tex][tex]\rightarrow125,000-25y+40y=149,000[/tex][tex]\rightarrow125,000+15y=149,000[/tex][tex]\rightarrow15y=24,000[/tex][tex]\therefore y=1600.[/tex]Now that we have y, we now solve for x to get:
[tex]x=5000-1600[/tex][tex]x=3400.[/tex]Hence x = 3400 and y = 1600 and the correct statements are as follows.
The number of tickets for sale at $25 should be 3400.
The number of tickets for sale at $40 should be 1600.