Answer:
45
Step-by-step explanation:
negative times negative equals positive
Let f be the function defined by f(x)=cx−5x^2/2x^2+ax+b, where a, b, and c are constants. The graph of f has a vertical asymptote at x=1, and f has a removable discontinuity at x=−2. (a) Show that a=2 and b=−4. (b) Find the value of c. Justify your answer. (c) To make f continuous at x=−2, f(−2) should be defined as what value? Justify your answer. (d) Write an equation for the horizontal asymptote to the graph of f. Show the work that leads to your answer.
Answer:
a) [tex]a = 2[/tex] and [tex]b = -4[/tex], b) [tex]c = -10[/tex], c) [tex]f(-2) = -\frac{5}{3}[/tex], d) [tex]y = -\frac{5}{2}[/tex].
Step-by-step explanation:
a) After we read the statement carefully, we find that rational-polyomic function has the following characteristics:
1) A root of the polynomial at numerator is -2. (Removable discontinuity)
2) Roots of the polynomial at denominator are 1 and -2, respectively. (Vertical asymptote and removable discontinuity.
We analyze each polynomial by factorization and direct comparison to determine the values of [tex]a[/tex], [tex]b[/tex] and [tex]c[/tex].
Denominator
i) [tex](x+2)\cdot (x-1) = 0[/tex] Given
ii) [tex]x^{2} + x-2 = 0[/tex] Factorization
iii) [tex]2\cdot x^{2}+2\cdot x -4 = 0[/tex] Compatibility with multiplication/Cancellative Property/Result
After a quick comparison, we conclude that [tex]a = 2[/tex] and [tex]b = -4[/tex]
b) The numerator is analyzed by applying the same approached of the previous item:
Numerator
i) [tex]c\cdot x - 5\cdot x^{2} = 0[/tex] Given
ii) [tex]x \cdot (c-5\cdot x) = 0[/tex] Distributive Property
iii) [tex](-5\cdot x)\cdot \left(x-\frac{c}{5}\right)=0[/tex] Distributive and Associative Properties/[tex](-a)\cdot b = -a\cdot b[/tex]/Result
As we know, this polynomial has [tex]x = -2[/tex] as one of its roots and therefore, the following identity must be met:
i) [tex]\left(x -\frac{c}{5}\right) = (x+2)[/tex] Given
ii) [tex]\frac{c}{5} = -2[/tex] Compatibility with addition/Modulative property/Existence of additive inverse.
iii) [tex]c = -10[/tex] Definition of division/Existence of multiplicative inverse/Compatibility with multiplication/Modulative property/Result
The value of [tex]c[/tex] is -10.
c) We can rewrite the rational function as:
[tex]f(x) = \frac{(-5\cdot x)\cdot \left(x+2 \right)}{2\cdot (x+2)\cdot (x-1)}[/tex]
After eliminating the removable discontinuity, the function becomes:
[tex]f(x) = -\frac{5}{2}\cdot \left(\frac{x}{x-1}\right)[/tex]
At [tex]x = -2[/tex], we find that [tex]f(-2)[/tex] is:
[tex]f(-2) = -\frac{5}{2}\cdot \left[\frac{(-2)}{(-2)-1} \right][/tex]
[tex]f(-2) = -\frac{5}{3}[/tex]
d) The value of the horizontal asympote is equal to the limit of the rational function tending toward [tex]\pm \infty[/tex]. That is:
[tex]y = \lim_{x \to \pm\infty} \frac{-10\cdot x-5\cdot x^{2}}{2\cdot x^{2}+2\cdot x -4}[/tex] Given
[tex]y = \lim_{x \to \infty} \left[\left(\frac{-10\cdot x-5\cdot x^{2}}{2\cdot x^{2}+2\cdot x-4}\right)\cdot 1\right][/tex] Modulative Property
[tex]y = \lim_{x \to \infty} \left[\left(\frac{-10\cdot x-5\cdot x^{2}}{2\cdot x^{2}+2\cdot x-4}\right)\cdot \left(\frac{x^{2}}{x^{2}} \right)\right][/tex] Existence of Multiplicative Inverse/Definition of Division
[tex]y = \lim_{x \to \pm \infty} \left(\frac{\frac{-10\cdot x-5\cdot x^{2}}{x^{2}} }{\frac{2\cdot x^{2}+2\cdot x -4}{x^{2}} } \right)[/tex] [tex]\frac{\frac{x}{y} }{\frac{w}{z} } = \frac{x\cdot z}{y\cdot w}[/tex]
[tex]y = \lim_{x \to \pm \infty} \left(\frac{-\frac{10}{x}-5 }{2+\frac{2}{x}-\frac{4}{x^{2}} } \right)[/tex] [tex]\frac{x}{y} + \frac{z}{y} = \frac{x+z}{y}[/tex]/[tex]x^{m}\cdot x^{n} = x^{m+n}[/tex]
[tex]y = -\frac{5}{2}[/tex] Limit properties/[tex]\lim_{x \to \pm \infty} \frac{1}{x^{n}} = 0[/tex], for [tex]n \geq 1[/tex]
The horizontal asymptote to the graph of f is [tex]y = -\frac{5}{2}[/tex].
Using asymptote concepts, it is found that:
a) Building a quadratic equation with leading coefficient 2 and roots 1 and -2, it is found that a = 2, b = -4.
b) c = -10, since the discontinuity at x = -2 is removable, the numerator is 0 when x = -2.
c) Simplifying the function, it is found that at [tex]x = -2, f(x) = -\frac{5}{3}[/tex].
d) The equation for the horizontal asymptote to the graph of f is [tex]y = -\frac{5}{2}[/tex]
-------------------------
Item a:
Vertical asymptote at [tex]x = 1[/tex] and discontinuity at [tex]x = -2[/tex] means that the the roots of the quadratic function at the denominator are [tex]x = 1[/tex] and [tex]x = -2[/tex].The leading coefficient is given as 2, thus, we build the equation to find coefficients a and b.[tex]2(x - 1)(x - (-2)) = 2(x - 1)(x + 2) = 2(x^2 + x - 2) = 2x^2 + 2x - 4[/tex]
[tex]2x^2 + ax + b = 2x^2 - 2x - 4[/tex]
Thus a = 2, b = -4.
-------------------------
Item b:
Removable discontinuity at [tex]x = -2[/tex] means that the numerator when [tex]x = -2[/tex] is 0, thus:[tex]-2c - 5(-2)^2 = 0[/tex]
[tex]-2c - 20 = 0[/tex]
[tex]2c = -20[/tex]
[tex]c = -\frac{20}{2}[/tex]
[tex]c = -10[/tex]
-------------------------
Item c:
With the coefficients, the function is:
[tex]f(x) = \frac{-10x - 5x^2}{2x^2 + 2x - 4} = \frac{-5x(x + 2)}{2(x - 1)(x + 2)} = -\frac{5x}{2(x - 1)}[/tex]
At x = -2:
[tex]-\frac{5(-2)}{2(-2 - 1)} = -\frac{-10}{-6} = -(\frac{5}{3}) = -\frac{5}{3}[/tex]
Thus, simplifying the function, it is found that at [tex]x = -2, f(x) = -\frac{5}{3}[/tex]
-------------------------
Item d:
The horizontal asymptote of a function is:
[tex]y = \lim_{x \rightarrow \infty} f(x)[/tex]
Thus:
[tex]y = \lim_{x \rightarrow \infty} \frac{-10x - 5x^2}{2x^2 + 2x - 4} = \lim_{x \rightarrow \infty} \frac{-5x^2}{2x^2} = \lim_{x \rightarrow \infty} -\frac{5}{2} = -\frac{5}{2}[/tex]
The equation for the horizontal asymptote to the graph of f is [tex]y = -\frac{5}{2}[/tex]
A similar problem is given at https://brainly.com/question/23535769
3.75 +(-4)+(-5.25) =
Answer:
-5.5
Step-by-step explanation:
Frank makes x dollars per week. His dad makes 4 times as much money as him each week. Each week their income adds up to 1.000 dollars. How much does frank earn each week? How much does franks dad earn each week?
Answer:
Frank earns $200.
His dad earns $800.
Step-by-step explanation:
Giving the following information:
His dad makes 4 times as much money as him each week. Each week their income adds up to 1.000 dollars.
We know that:
x= Franks income
4x= Dads income
y= total income
y= x + 4x
1,000= 5x
1,000/5=x
200=x
Frank earns $200.
His dad earns $800.
3/2(4x – 1) – 3x = 5/4 – (x + 2
Answer:
If you need to evaluate the equation, the answer would be x=19/16.
Step-by-step explanation:
If a = 3 and b = 4, then find the value of 2a + 3b A. 15 B. 16 C. 17 D. 18
Answer:
18
Step-by-step explanation:
2a+3b
2•3+3•4
6+12
18
Answer:
2124+272cd
Step-by-step explanation:
Sarah is working as a truck driver. She gets paid $100 everyday and then $22 per hr driving. How many hours does sarah have to work to make $276 in one day
Answer:8
Step-by-step explanation: she makes $100 a day so you would subtract that from the total (276) then you would divide the remainder (176) by the amount she makes an hour (22) then you would end up with 8 hours
Answer:
8 hours
Step-by-step explanation:
For $22=1 hour
For $1=1/22 hours
For $176=176/22 hours=8 hours
And she gets also $100 extra
So for $276 she works 8 hours.
Please answer asapWhat is the measure of an exterior angle of a regular 13-sided polygon? Enter your answer as a decimal in the box. Round to the nearest tenth of a degree.
Answer:
Step-by-step explanation:
it is given by the formula=4× 90/ n
where n = number of sides
exterior angle= 4 x 90/13
=360/13=[tex]27.7[/tex]°
2 over 3 = x-1 all over 4
Answer:
x = [tex]\frac{11}{3}[/tex]
Step-by-step explanation:
Given
[tex]\frac{2}{3}[/tex] = [tex]\frac{x-1}{4}[/tex] ( cross- multiply )
3(x - 1) = 8
3x - 3 = 8 ( add 3 to both sides )
3x = 11 ( divide both sides by 3 )
x = [tex]\frac{11}{3}[/tex]
which of the following sets represents a function
Answer:
the top one
Step-by-step explanation:
one input cannot have more than one output
why and how do you do this I am having a hard time
Answer:
Peter walks at a rate of 13/4 miles per hour.
Step-by-step explanation:
Peter walks at a rate of 13/4 miles per hour because every hour, represented by x, he walks 3.25 hours or 13/4. y is the total miles he walked so if he walks two hours the y would equal 6.5 or 26/4 and the x would equal 2.
solve x 3(x+1)=-2(x-1)-4
Answer:
x= -1/3 or -2
Step-by-step explanation:
3x^2 +3x= -2x +2-4
3x^2 + 5x +2=0
by factorization method
3x(x+2) 1(x+2)=0
(3x+1)(x+2)
Answer:
x3(x+1)= –2(x-1)–4
3x^2+3x=–2x+2-4
3x^2+3x+2x=2-4
3x^2+5x=–2
3x^2+5x+2=0
3x^2+3x+2x+2=0
3x(x+1)2(x+1)=0
(3x+2)(x+1)=0
3x+2=0 or x+1=0
3x=–2 or x=–1
x=–2/3 or –1
Step-by-step explanation:
–remove brackets by multiplication
–form a quadratic equation
–and then solve either by means of quadratic formula or factorisation method
*PLEASE ANSWER, NEED HELP* Which of the following is not a type of probability model? a.) A systematic list b.) An area model c.) A spinner d.) A tree diagram
Answer:
I think a spinner is not a type of probability model.
I am not sure tho
Option (c) A spinner is not a probability model
What is Probability model?A probability model is a mathematical representation of a random phenomenon.
Here,
In systematic list, the outcomes for an event can be listed in an organized or systematic way to make sure that none of the possible outcomes is missed out.
In area model, area models can be used to represent simple probabilities.
A probability tree diagram is used to represent the probability of occurrence of events without using complicated formulas.
But spinner is not a probability model.
Hence, option (D) Spinner is not a probability model
Learn more about Probability model here
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☆ please help! I got an answer but I think its wrong ☆
Answer:
-3,5,-10,16
Step-by-step explanation:
solve for y: 3x - y = 4
Answer:
y=12x/5
x=5y/12
Step-by-step explanation:
y:3x-y=4
y=4(3x-y)
because if there is divide sign if it will go to another side it will be multiply
y=12x-4y
y+4y=12x
5y=12x
y=12x/5
or, x=5y/12
Hope it is helpful for you
if it is please follow me
How can you determine what type of solution an equation has when solving? Explain.
Answer: Answer is in the steps
Step-by-step explanation:
If the system of equations have the same slope and same y-intercepts then it means that they have infinitely many solutions.
if the system of equations have the same slope but different y-intercepts it means that they have no solution because they will never intersect but will always form parallel lines.
If the system of equations have different slopes and different y-intercepts or have same y-intercepts and different slopes it means that they have exactly on solution.
which ordered pair is the solution of the equation? -3x+4y=14
Answer:
For Khan: A-Only (2,5)
Step-by-step explanation:
Is 41/50 closer to 9/11 or 10/11? Verify your answer
Answer: No it is closer to 9/11
Step-by-step explanation:
Answer: It is closer to 9/11
Step-by-step explanation: 41/50 in decimal form is 0.82.
9/11 in decimal form is 0.8181, which rounds up to 0.82
10/11 in decimal for is 0.9090, which rounds up to 0.91.
So,
41/50 = 0.82
9/11 =0.8181 OR 0.82
10/11 = 0.9090 OR 0.91
please answer ASAP ......
5/k = 15/20 what would the k be?
Answer:
20/3
Step-by-step explanation:
5/k=15/20
5/k=3/4
5 x 4/3 = k
k=20/3
What integer describes gaining 14 pounds ?
Answer:
14
Step-by-step explanation:
Since gaining fourteen pounds means going in a positive direction, we can say the gain of fourteen pounds numerically as 14.
What type of proof is used extensively in Geometry?
Answer: two-column
Step-by-step explanation:
Answer:
two-column proof
Step-by-step explanation:
quizlet
A sandwich shop employee
named Pedro takes 5.7
minutes to make a sandwich.
How long does it take him to
make 6 sandwiches?
Let x, y, z be numbers. (x2 yz4)3 =
If m < A = 4x + 12 and x = 15, is < A acute, obtuse, or right? Justify your answer.
Step-by-step explanation:
Angle A = 4x + 12 = 4(15) + 12 = 72°.
Acute is for angles that are less than 90°.
Obtuse is for angles that are between 90° and 180°, not inclusive.
Right is for angles that are 90°
Hence the answer is acute.
The angle is an acute angle
An acute angle is an angle that is less than 90 degrees
An obtuse angle is an angle that is greater than 90 degrees and less than 180 degrees
In order to determine which type of angle it is, solve for angle A
4x + 12
4(15) + 12
60 + 12 = 72
The angle is acute because it is less than 90 degrees
Please check the attached image for an image of an acute angle
To learn more about acute angles, please check: https://brainly.com/question/4022946?referrer=searchResults
what is Twice x, plus 8 , is the same as -15?
Answer:
x=11.5
Step-by-step explanation:
2x+8=-15
x=-23/2
x=-11.5
Answer:
= -11.5
Step-by-step explanation:
2x + 8 = - 15
2x = -15-8
2x = - 23
divide both sideby 2
2x = -23
2 2
x = -11.5
What is the equation of a line with a slope of –2 that passes through the point (6, 8)?
Answer:
The answer is
y = - 2x + 20Step-by-step explanation:
To find the equation of the line given a point and slope we use the formula
[tex]y - y_{1} = m(x - x _{1})[/tex]
where
m is the slope
( x1 , y1) is the point
From the question
The point is (6, 8) and
slope - 2
The equation of the line is
[tex]y - 8 = - 2(x - 6) \\ y - 8 = - 2x + 12 \\ y = - 2x + 12 + 8[/tex]
We have the final answer as
y = - 2x + 20Hope this helps you
Answer:
C on edge
Step-by-step explanation:
just took the quiz
Alex’s house (point F) lies on the same street as her school (point H). Alex’s bus stop (point G) lies between her house and her school.
Given FG = (2x) meters, GH = 1,000 meters, and FH = 1,200 meters, what is x?
The value of x in the expression will be 100.
What is a line segment?A line segment in geometry is a section of a line that has two clearly defined endpoints and contains every point on the line that lies within its confines.
Given that FG = (2x) meters, GH = 1,000 meters, and FH = 1,200 meters,
The value of x will be calculated by making an equation for the line segment below:-
FG + GH = FH
2X+1000=1200
2X = 1200 - 1000
2X = 200
X = 100 meters
2X + 1000= 1200
Therefore, the value of x in the expression will be 100.
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−0.75 − (−
5
2
)+0.4+(−
4
3
)
Answer:
-13
Step-by-step explanation:
PLEASE HELP, ITS TIMED SO HURRY PLEASE>>>>>>>
Answer:
The answer is noncollinear
Step-by-step explanation:
Answer:
Non-collinear because the points dont line up to be colinnear. learned this today dead a** its not hard. pay attention
Step-by-step explanation:
Which point is located in quadrant IV?