Answers
Answer:
Domain: (-∞, -1) ∪ (-1, 3) ∪ (3, ∞)Range: [0, ∞)Continuity: Point discontinuity (hole) at x = 3.Infinite discontinuity (vertical asymptote) at x = -1.NoneDecreasing function: (-∞, -1) ∪ (-1, ∞)Symmetry: Neither
Step-by-step explanation:
Given function:
[tex]y=\dfrac{x-3}{x^2-2x-3}[/tex]
DomainThe domain is the set of all possible input values (x-values).
Factor the denominator of the function:
[tex]\implies y=\dfrac{x-3}{(x-3)(x+1)}[/tex]
The domain of the given function is restricted as the function is undefined when the denominator equals zero, and when both the numerator and denominator equal zero.
Therefore, the function is undefined when x = -1 and x = 3.
Therefore, the domain is (-∞, -1) ∪ (-1, 3) ∪ (3, ∞).
RangeThe range is the set of all possible output values (y-values).
As the domain is restricted, the range is also restricted.
Therefore, the range is (-∞, 0) ∪ (0, ¹/₄) ∪ (¹/₄, ∞).
ContinuityA function f(x) is continuous when, for every value [tex]c[/tex] in its domain:
[tex]\text{$f(c)$ is de\:\!fined \quad and \quad $\displaystyle \lim_{x \to c} f(x) = f(c)$}[/tex]
Point Discontinuity (holes): When a rational function has a factor with an x that is in both the numerator and the denominator.
Therefore, there is a point discontinuity at x = 3.
Infinite Discontinuity: A vertical asymptote occurs when the denominator of a rational function is zero. This is considered an infinite discontinuity when the graph exists on both sides of the asymptote.
Therefore, there is an infinite discontinuity at x = -1.
Maximums and MinimumsStationary points occur when the gradient of a graph is zero.
Therefore, to find the x-coordinate(s) of the stationary points of a function, differentiate the function, set it to zero and solve for x.
[tex]\begin{aligned}y & =\dfrac{x-3}{x^2-2x-3}\\\\\implies \dfrac{\text{d}y}{\text{d}x} & = \dfrac{(x^2-2x-3)(1) - (x-3)(2x-2)}{\left(x^2-2x-3\right)^2}\\\\ & = \dfrac{(x-3)(x+1)-(x-3)(2x-2)}{\left((x-3)(x+1)\right)^2}\\\\& = \dfrac{(x-3)[(x+1)-(2x-2)]}{(x-3)^2(x+1)^2}\\\\ & = \dfrac{(x-3)(-x+3)}{(x-3)^2(x+1)^2}\\\\ & = \dfrac{-(x-3)^2}{(x-3)^2(x+1)^2}\\\\& = \dfrac{-1}{(x+1)^2}\end{aligned}[/tex]
[tex]\begin{aligned}\dfrac{\text{d}y}{\text{d}x} & = 0\\\implies \dfrac{-1}{(x+1)^2} & = 0\\ -1 & \neq 0\end{aligned}[/tex]
Therefore, there are no stationary points and no minimum/maximum points.
Increasing/Decreasing Function[tex]\textsf{A function is \textbf{increasing} when the \underline{gradient is positive}} \implies \dfrac{\text{d}y}{\text{d}x} > 0[/tex]
[tex]\textsf{A function is \textbf{decreasing} when the \underline{gradient is negative}} \implies \dfrac{\text{d}y}{\text{d}x} < 0[/tex]
Increasing
[tex]\begin{aligned}\dfrac{\text{d}y}{\text{d}x} & > 0\\\implies \dfrac{-1}{(x+1)^2} & > 0\\ \dfrac{1}{(x+1)^2} & < 0\\ \textsf{If $\dfrac{1}{a} < 0$ then $a < 0$}:\\(x+1)^2 & < 0\\ \textsf{As $(x+1)^2\geq 0$, no solution.}\end{aligned}[/tex]
Decreasing
[tex]\begin{aligned}\dfrac{\text{d}y}{\text{d}x} & < 0\\\implies \dfrac{-1}{(x+1)^2} & < 0\\ \dfrac{1}{(x+1)^2} & > 0\\ \textsf{If $\dfrac{1}{a} > 0$ then $a > 0$}:\\(x+1)^2 & > 0\\ \end{aligned}[/tex]
[tex]\textsf{For $a^n > 0$, if $n$ is even then $a < 0$ or $a > 0$}[/tex]
[tex]\implies x+1 < 0 \implies x < -1[/tex]
[tex]\implies x+1 > 0 \implies x > -1[/tex]
Combine the intervals:
(-∞, -1) ∪ (-1, ∞)
Therefore:
The function is decreasing in the interval (-∞, -1) ∪ (-1, ∞).Symmetry[tex]\textsf{A function is \underline{even} when $f(x) = f(-x)$ for all $x$.}[/tex]
[tex]\textsf{A function is \underline{odd} when $-f(x) = f(-x)$ for all $x$.}[/tex]
As there is no symmetry about the y-axis (even symmetry) and no origin symmetry (odd symmetry), the symmetry of the function is neither.
Answer:
Domain: (-∞, -1) ∪ (-1, 3) ∪ (3, ∞)
Range: [0, ∞)
Continuity: Point discontinuity (hole) at x = 3.
Infinite discontinuity (vertical asymptote) at x = -1.
None
Decreasing function: (-∞, -1) ∪ (-1, ∞)
Symmetry: Neither
Related Questions
Express 6 hours 28 minutes and 16 seconds in decimal form. Round your answer to two decimal places.
Answers
Answer: 6.47 hours
Step-by-step explanation:
First, you need to change your seconds into terms of minutes.
16 seconds / 60 s in a min = .266 min
Add that to your minutes:
28+.266 = 28.266 minutes
Now convert your minutes into hours
28.266 min/60 min in an hour=.47
Add that to your hours:
6+.47= 6.47 hours total
8. How much energy does a 12 V battery in your car use to move 2.5 C through the electrical
circuit?
Answers
The energy required is 30 watts second.
In order to accomplish work and to produce heat and light, energy must be transferred to a body or to a physical system. Energy is the quantitative attribute that does this. Energy is a preserved resource; according to the rule of conservation of energy, energy can only be transformed from one form to another and cannot be created or eliminated.
The voltage of a battery is 12 V and;
The charge is 2.5C.
Now, energy can be defined as the product of voltage and charge.
Therefore,
E = V × C
E = 12 × 2.5
E = 30 watt second
Hence, the energy needed by the 12 V battery to move 2.5 C through the electrical circuit is 30 watts second.
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10-0.5(12+8b)+b
pls help if you can!
Answers
Answer: 4-3b
Step-by-step explanation:
10-0.5(12+8b)+b First, distribute -0.5 into 12 and 8b
10-6-4b+b Then combine like terms
4-3b Done!
The ratio 4:5 is equivalent to
(i) 80%
(ii) 60%
(iii) 40%
(iv) 20%
If u ans i will give 40 points and mark brainlyest
Answers
Answer:
(i)
Step-by-step explanation:
express the ratio as a fraction, that is
4 : 5 = [tex]\frac{4}{5}[/tex]
to convert the fraction to a percentage multiply it by 100%
[tex]\frac{4}{5}[/tex] × 100% = 4 × 20% = 80%
Xaivier played video games for 1 hour and 45 minutes before he went to bed. If he went to bed at 9:00 pm,what time did he start playing video games
Answers
Answer:7:15 pm
Step-by-step explanation:
We know he went to bed at 9:00 PM and spent 1 hr and 45 minutes playing games. Start by subtracting one hour away, so we get 8:00 PM. subtract 45 minutes from that and he would have started playing at 7:15 PM.
How many milliliters of a 28% sugar Solution do we need to mix with 75 milliliters of a 10% sugar solution so that the resulting mixture is a 13% sugar solution?
Answers
We need to mix 15 ml of 28 % of solution to get the resulting mixture as a 13 % sugar mixture.
Let the amount of 28 % sugar solution be x ml.
We are given that:
10 % sugar solution = 75 ml
Total solution = 75 ml + x ml
We need the solution to be 13 % sugar solution.
So, we get that:
28 % x + 10 % ( 75 ) = 13 % ( 75 + x )
28 x + 750 = 13 ( 75 + x )
28 x + 750 = 975 + 13 x
28 x - 13 x = 975 - 750
15 x = 225
x = 225 / 15
x = 15
Therefore, we need to mix 15 ml of 28 % of solution to get the resulting mixture as a 13 % sugar mixture.
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Find the slope of each line
Answers
Line n= 5/2
Those 2 lines are parallel because they have same slope
Line m= 2/3
Line s= 2/3
Line m and s are parallel because they have same slope
Slope = rise/run
A person's systolic blood pressure, which is measured in millimeters of mercury (mm Hg), depends on a person's age, in years. The equation
p=0.007y^2-0.003y+122
gives a person's blood pressure, p,at the age y year
A.) Find the systolic pressure, to the nearest tenth of a millimeter, for a person of age 41 years.
Answers
The systolic pressure, to the nearest tenth of a millimeter, for a person of age 41 years is; 133.6 mm Hg
How to solve problems using quadratic equations?We are given the equation that gives a person's blood pressure, p at the age y year is;
p = 0.007y² - 0.003y + 122
To find the systolic pressure for a person of age 41 years, just plug in y = 41 into the given quadratic equation to get;
p(41) = 0.007(41)² - 0.003(41) + 122
p(41) = 11.767 - 0.123 + 122
p(41) = 133.6
Thus, the systolic pressure, to the nearest tenth of a millimeter, for a person of age 41 years is; 133.6
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Which expression is equivalent to the given expression?
2n−6
A −2(n−3)
B 6−2n
C −4n
D 2(n−3)
Answers
For the universal set, u= (5, 6, 7, 8, 9) , complete the parts below.
Write your answers in roster form or as .
Answers
Given the subset A, the complement of A, A' is {6}
Given the complement of C, C is {5, 7, 9}
What are the answers in roster form?The universal set contains all the elements of the subsets. ' means complement = it means elements that are not in the subset but in the universal set.
To write a set in a roster form, the elements in the set are written in a row within curly brackets - { }. The elements in the set are then separated by commas.
A' = {6}
C = {5, 7, 9}
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what is? 9x-x-2=3(2x-3)+ax
Answers
The perimeter of any rectangle in which the length is 4 more than twice the width is P = 6 w + 8 , where w is the width. Which formula can be used to find the width given the perimeter? Multiple choice question. cross out A) w = P − 4 3 cross out B) w = 1 6 P − 8 cross out C) w = − P + 8 6 cross out D) w = P − 8 6
Answers
Answer: option D
Step-by-step explanation:
Given equation:
P = 6 w + 8 [eq1]
l=2w+4 [where l is length]
using eq1 we have:
6w=P-8
w=(P-8)/6
hence option D
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Please help!
Simplify:
Answers
Step-by-step explanation:
[tex] = ( \frac{3}{5} {)}^{2} [/tex]
[tex] = \frac{ {3}^{2} = 3 × 3 }{ {5}^{2} = 5 × 5 } \\ [/tex]
[tex] = \frac{9}{25} \\ [/tex]
☘️☘️☘️...
Answer:
[tex] \frac{9}{25} [/tex]
Step-by-step explanation:
[tex] { ( \frac{3}{5} )}^{2} \\ = ( \frac{ {3}^{2} }{ {5}^{2} } ) \\ = \frac{9}{25} [/tex]
Given a graph for the transformation of f(x) in the format g(x) = f(x) + k, determine the k value.
Ok = -3
Ok = 1
Ok = 4
Ok=5
-8 -7
f(x)
-6 -5
y¹¹
-10
g(x)
4-
-3 -2 -1 0
-4-
-2
-3
-4
-5
Answers
Answer:
4
Step-by-step explanation:
g(x) is the graph of f(x) shifted up 4 units.
A bowl weighs 11/40 pound.
a. Express this weight as a decimal.
b. Explain how you can find the decimal by thinking of the denominator as a 4 and using mental math.
Answers
Answer: 0.275
Step-by-step explanation:
a. To express 11/40 as a decimal, you need to divide 11 by 40. (Steps shown in image)
b. Sorry, I can't figure out the answer to this. I don't recommend making the denominator a 4 as the numerator would become 1.1.
If you have the function f(x) = 5^x - 5 then find the value of f(t)
Answers
The value of the function is [tex]5^t-5[/tex] .
A function is defined as a expression containing one or more variables.
A function, in technical terms, is a mechanism to connect a collection of inputs to a set of outputs.
A function of a single variable (say x) is defined mathematically as g(x).
Let us consider a function of a variable x such that it is defined as:
g(x)=x+1
Here the output are calculated for all values of x . At x=1 the value of the function is denoted as g(1)=1+1=2.
Mathematically to calculate the value of the function f(x) for any particular value of x we substitute the value of x into the value of f(x).
Now the given function is given as:
[tex]f(x)=5^x-5[/tex]
Therefore at x=t,
[tex]f(t)=5^t-5[/tex]
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Find the unique solution to this system of equations. Give your answer as a point
Answers
Answer:
(x, y, z) = (-4, -5, 2)
Step-by-step explanation:
You want the solution to the 3×3 system of equations ...
3x -y -4z = -15-4x -4y -z = 342x +3y +5z = -13ToolsThere are many tools available for solving a system of equations. A typical hand-held graphing calculator offers at least two methods. An online graphing calculator solution is shown in the attachment.
For the attached solution, we defined z in terms of x and y using the second equation. This effectively reduces the system to a 2×2 system, which is easily solved. The same "substitution" method can be used when solving the system by hand.
The graph shows the solution to be (x, y, z) = (-4, -5, 2).
Manual solutionUsing z = -4x -4y -34 from the second equation, the system reduces to ...
3x -y -4(-4x -4y -34) = -15
19x +15y = -151
and
2x +3y +5(-4x -4y -34) = -13
-18x -17y = 157
Writing these equations in general form, we have ...
19x +15y +151 = 018x +17y +157 = 0The "cross multiplication method" gives the solution for x and y as ...
D = (19)(17) -(18)(15) = 53
x = (15(157) -17(151))/D = -212/53 = -4
y = (151(18) -157(19))/D = -265/53 = -5
Then z is ...
z = -4(-4) -4(-5) -34 = 16 +20 -34 = 2
The solution is (x, y, z) = (-4, -5, 2).
__
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Lizzy had #500, John has #50
FIFI will have #700.Who has more money amongst the three.
Answers
Answer:
John
Step-by-step explanation:
it says Lizzy had which means we don’t know how much she has now. It says Fifi will have so we don’t know how much they have right now. But it says John has so we know what he has so therefore he has the most as of right now
According to a recent study, the number of prisoners in a prison decreased by 312 inmates over a one-year period. If the decrease was the
same for each of the 12 months, write the monthly decrease as a quotient of integers and find the monthly decrease.
The expression to find the monthly decrease in prisoners is.
(Do not simplify.)
The number of inmates in the prison decreased by inmates per month.
(Simplify your answer.)
Answers
Over a year, a prison's population dropped by 312 inmates. We know that 1 Year = 12 Months.
Additionally, considering that each of the 12 months declined at the same rate.
the Monthly decrease in members = 312 / 12 = 26
Therefore, the expression used to calculate the monthly drop in prisoner population = Number of prisoners decreased per year / Number of months
Note: If the monthly drop rate is the same, the aforementioned equation is correct ( for all months in that year)
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5(x+4) =1/2(12) (2X-6) STEPS TO SOLVE
Answers
hi
5(x+4) =1/2 *(12)* (2X-6)
5x+20 = 6 * ( 2x-6)
5x+20 = 12x-36
5x-12x = -36-20
-7x = -56
x = -56/-7
x = 8
Market research was undertaken to determine whether an ad campaign was successful for the software company. 15 men and 75 women were asked if they had seen the company’s latest TV advertisement. Altogether, 2 5 of the people said ‘yes’. 6 15 of the men said ‘yes’. What percentage of women said ‘no’ out of the total survey population?
Answers
50% of women who said ‘no’ out of the total survey population if 15 men and 75 women were asked if they had seen the company’s latest TV advertisement.
What is the percentage?It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.
It is given that:
Market research was undertaken to determine whether an ad campaign was successful for the software company.
Total number of people = 15 + 75 = 90
The number of people who said yes = (2/5)90 = 36
The number of people who said no = 90 - 36 = 54
The number of men said yes = (6/15)x15 = 6
The number of men said No = 15 - 6 = 9
The number of women who said no = 54 - 9 = 45
Percentage = (45/90)x100
The percentage of women who said ‘no’ out of the total survey population:
= 50%
Thus, 50% of women who said ‘no’ out of the total survey population if 15 men and 75 women were asked if they had seen the company’s latest TV advertisement.
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10.
An airplane has economy and first class seating.
There are s seats in each row in economy.
There are t seats in each row in first class.
There are 9 rows in first class and 24 rows in economy.
Write down an expression, in terms of s and t, for the number of seats on the
airplane.
Answers
The expression, in terms of s and t, for the number of seats on the airplane is given by 24 s + 9 t.
We are given that there are 2 types of seating in an airplane which are economy and first class seating.
Also,
We are given that:
There are s seats in economy in each row and there are t seats in first class in each row.
Now, there are 24 rows in economy class seating.
So, number of seats in economy class = 24 × s = 24 s
And, there are 9 rows in first class seating.
So, number of seats in first class = 9 × t = 9 t
The total number of seats then will be = 24 s + 9 t
Therefore the expression, in terms of s and t, for the number of seats on the airplane is given by 24 s + 9 t.
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Which is the least whole number that rounds to 2000 when rounded to the nearest hundred
Answers
The required number is 2001.
To solve this problem we need to know the concept of rounding off.
Rounding off a number means making the value of number equal to the nearest tens place.
When we round off 45 we will get 50; when we round off 44 we get 40.
So, if a number has 5 at its unit place then it will be rounded off to the tens place which is ahead of 5 and if a number is less than 5 then it is rounded off to the tens place before that number.
Here it is asked to find the least whole number that rounds off to 2000 when rounded off to nearest hundred which will be 2001.
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What is the midpoint of the line segment from p1 to p2 is (-4,1). If P1 =(-4,5) what is p2??
Answers
Find the distance between P1 and the midpoint:
P1 - midpoint
(-4,6) - (-4, 1)
Start with the "x" coordinates. Since the "x" coordinate is the same (x = -4) for both P1 and the midpoint, we know the line is vertical. P2's
"x" coordinate will also equal -4.
Next subtract the "y" coordinates:
P1 - midpoint
6 - 1 = 5
The distance between P1 and the midpoint is 5 coordinate points.
Next subtract 5 from the midpoint's "y" coordinate to find P2.
midpoint - 5
1 - 5 = -4.
P2 = (-4, -4)
A flight averages 460 miles per hour. The return flight averages 500 mph because of a tailwind. The total flying time is 4 hours and 48 minutes. How long is each flight
Answers
The first flight is 2 hours and second flight is 2 hours and 48 minutes
The two distances are the same (out and back),
so set them equal.
That distance is cover by having one is (rate)(time) equal to other is (rate)(time).
Let,
One time is “x” and the other is “4.8-x.”
One average rate is 460 per hour
and the other average rate is 500 per hour.
we know , Speed = [tex]\frac{distance}{time}[/tex]
so , distance = speed × time
put in this formula we get the value of X
⇒460 x = 500 (4.8 -x)
⇒460 x = 2400 - 500x
⇒900 x = 2400
⇒x = 2.5 hours for the slower plane.
⇒4.8- x = 2.3 hours for the faster plane.
The formula speed = distance ÷ time can be rearranged, just like any other equation.
Here detail explanation about relation between speed distance and time :
The formula can be rearranged in three ways:
speed = distance ÷ time
distance = speed × time
time = distance ÷ speed
To calculate one of the variables (speed, distance or time) we need the other two.
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PLEASE HELP ME FAST
An article claimed the state's typical price of a gallon of gasoline is $2.69. The prices from 5 gas stations surveyed were $2.79, $2.99, $2.69, $2.89, and $2.69.
According to this data, what is wrong with the article's statement?
Answers
This statement in the article is wrong because $2.69 is not the mean price ($2.81)
How to determine what is wrong with the data?The prices from 5 gas stations surveyed are given as:
$2.79, $2.99, $2.69, $2.89, and $2.69.
There are no outliers in the dataset
So, the dataset can be summarized by the mean
The mean is
Mean = Sum/Count
So, we have
Mean = ($2.79+ $2.99+ $2.69+ $2.89+$2.69)/5
Evaluate
Mean = 2.81
The article quoted $2.69 as the typical price
This statement in the article is wrong because $2.69 is not the mean price ($2.81)
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Three rational numbers between 1/3 and 1/5
Answers
Three rational numbers between
1/3 and 1/5 are 4/15, 3/10, and 7/30
Given:The two numbers are 1/3 and 1/5
To find:The rational number between two numbers
Solution:Now, to find the rational number, add the numbers and divide their sum by 2.
(i) First rational number between
1/3, 1/5 = 1/3+1/5=8/5
Divide 18/5 by 2------> 8/15 x 1/2= 8/30= 4/15
The series becomes, 1/3, 4/15, 1/5
(ii) Second rational number
= 4/15+1/5 = 7/15
Divide
7/15 by 2 ------> 7/15 x 1/2 = 7/30
Hope this helps!
How do I find out the check amount before the tip when the total amount was $55 with a 10% tip
Answers
Answer:$49.50
Step-by-step explanation:
First you have to find what 10% of 55 is. So if you times 10% by 55 you come out with $5.5. So after that you must subtract 5.5 from 55 which comes out to $49.50.
Choose the best answer.
Use the conditional statement to answer the question.
If an angle measures 43 degrees, then the angle is acute.
Choose the best answer.
The statement is false, but the inverse is true.
Both the statement and its inverse are false.
Both the statement and its inverse are true.
The statement is true, but the inverse is false.
Answers
Answer:
D
Step-by-step explanation:
Acute angle are angles less than 90°.And 43° is less than 90° and the inverse is not less than 90°.
Maureen knows the domain of f(x) is the set of positive real numbers if she graphs f(x) on the coordinate plane what must be true about the graph
Answers
The correct option is C, if f(x) has a domain only with positive values, the graph will be on the first and fourth quadrant.
What must be true about the graph?We know that the domain of f(x) is only the set of positive real numbers. Now remember that the domain of a function is the set of the inputs of that function (the inputs are on the horizontal axis).
Then the graph of f(x) will only be on the positive side of the horizontal axis. We know that the two quadrants that are on that region are the first quadrant and the fourth quadrant.
Then if f(x) has a domain only with positive values, the graph will be on the first and fourth quadrant, this means that the correct option is C.
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