Answers
20. A rational function [tex]\frac{p(x)}{q(x)}[/tex] (where [tex]p,q[/tex] are polynomials in [tex]x[/tex]) has a removable discontinuity at [tex]x=a[/tex] if for some positive integer [tex]n[/tex] we can factorize
[tex]p(x) = (x-a)^n p^*(x)[/tex]
[tex]q(x) = (x-a)^n q^*(x)[/tex]
[tex]\implies \dfrac{p(x)}{q(x)} = \dfrac{(x-a)^n p^*(x)}{(x-a)^n q^*(x)} = \dfrac{p^*(x)}{q^*(x)}[/tex]
That is, we "remove" the discontinuity at [tex]x=a[/tex] because while [tex]x\neq a[/tex], we have [tex]\frac{x-a}{x-a}=1[/tex]. The discontinuity is still there, since [tex]\frac{p(a)}{q(a)}[/tex] is undefined, but in the limit sense we can essentially ignore the factors of [tex]x-a[/tex]. The graph of [tex]\frac{p(x)}{q(x)}[/tex] will have all the features of [tex]\frac{p^*(x)}{q^*(x)}[/tex] aside from a hole at [tex]x=a[/tex].
In this case, we have
[tex]f(x) = \dfrac{x+4}{x^2-x-20} = \dfrac{x+4}{(x+4)(x-5)}[/tex]
and when [tex]x\neq-4[/tex], we can cancel the factor of [tex]x+4[/tex] so that
[tex]f(x) = \begin{cases}\dfrac1{x-5} & \text{if }x \neq -4 \\\\ \text{DNE} & \text{if }x = -4\end{cases}[/tex]
("DNE" = "does not exist", i.e. is undefined. For some reason "undefined" wouldn't render...)
This means [tex]x=-4[/tex] is a removable discontinuity.
We cannot do the same with the factor of [tex]x-5[/tex], so in contrast [tex]x=5[/tex] is a non-removable discontinuity.
21. The pieces of [tex]f(x)[/tex] defined on [tex]x<2[/tex] and [tex]x>2[/tex] are themselves continuous since they are polynomials. Then the continuity of [tex]f(x)[/tex] over the entire real line depends on the point where the pieces meet.
Here we have
[tex]f(x) = \begin{cases}x+3 & \text{if }x\le2 \\ cx+6 & \text{if }x>2\end{cases}[/tex]
so the pieces meet at [tex]x=2[/tex]. Continuity at this point requires that the both limits from either side of [tex]x=2[/tex] be the same. This means
[tex]\displaystyle \lim_{x\to2^-} f(x) = \lim_{x\to2} (x+3) = 2+3 = 5[/tex]
[tex]\displaystyle \lim_{x\to2^+} f(x) = \lim_{x\to2} (cx+6) = 2c + 6[/tex]
Solve for [tex]c[/tex].
[tex]2c + 6 = 5 \implies 2c = -1 \implies \boxed{c = -\dfrac12}[/tex]
Related Questions
The original price of a mountain bike was reduced by $125 . if p= the mountain bike's original price in dollars, which algebraic expression represents the reduced price?
Answers
The algebraic expression that represents the reduced price is p - 125
How to determine the reduced price?The reduction is given as:
Reduction = $125
The original price is given as:
Original = P
The reduced price is calculated as:
Reduced price = original - reduction
This gives
Reduced price = p - 125
Hence, the algebraic expression that represents the reduced price is p - 125
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A cash card has a starting value of $25. If the card is not used within the first year of its purchase, the value on the card begins to decrease by $2.50 per month.
Answers
Using linear function concepts, we have that from the information given:
The relationship is linear.The independent variable is the number of months.The dependent variable is the value of the card.What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.x is the independent variable.y is the dependent variable.From the information given, the y-intercept is of 25 and the slope is of $2.50, hence the value after t months is given by:
V(t) = 25 - 2.5t.
Thus:
The relationship is linear.The independent variable is the number of months.The dependent variable is the value of the card.More can be learned about linear functions at https://brainly.com/question/24808124
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Question 12
You are negotiating to make a 7-year loan of $20,000 to Breck Inc. To repay you, Breck will pay $2,500 at the end of Year 1, $5,000 at
the end of Year 2, and $7,500 at the end of Year 3, plus a fixed but currently unspecified cash flow, X, at the end of each year from
Year 4 through Year 7. Breck is essentially riskless, so you are confident the payments will be made. You regard 8% as an
appropriate rate of return on a low risk but illiquid 7-year loan. What cash flow must the investment provide at the end of each of
the final 4 years, that is, what is X?
(A) $2,520.02
B) $2,831.49
$3,369.47
$2,491.71
0.5 Points
E $2,661.60
Answers
The required x = $2,831.49. Option B) is correct.
7-year loan of $20,000 to Breck Inc. To repay you, Breck will pay $2,500 at the end of Year 1, $5,000 at the end of Year 2, and $7,500 at the end of Year 3, plus a fixed but currently unspecified cash flow, X, at the end of each year from Year 4 through Year 7.
What is invesment?When some apply there money on something so that he can get higher value in returns of his money.
for the final 4 years
20000=2500/1.08+5000/(1.08^2)+7500/(1.08^3)+x/(1.08^4)+x/(1.08^5)+x/(1.08^6)+x/(1.08^7)
x = $2,831.49
Thus the required value of x = $2,831.49.
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The step function f(x) is graphed. -54-3 -2 -1 4 3 EXO 4 4 y 1 CHELLEREN 2 3 What is the value of f(0)? O -2 O -1 0 0 1
Answers
Answer: -2
Step-by-step explanation:
When x = 0, y = -2.
So, f(0) = -2.
Identify the algebraic expression for the following: the quotient of 12 and a number y increased by 7
Answers
The algebraic expression for the given word problem is 12/y + 7
Writing an expressionFrom the question, we are to write the algebraic expression for the given word problem
The given word problem is
"the quotient of 12 and a number y increased by 7"
First,
"the quotient of 12 and a number y" can be expressed as
12/y
Then,
"the quotient of 12 and a number y increased by 7" becomes
12/y + 7
Hence, the algebraic expression for the given word problem is 12/y + 7
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Emma read the statement “the quotient of six and a number, x, is the same as negative two times the difference of x and 4.5.” She translated it as shown below. 6 x = negative 2 x minus 4.5 What errors did Emma make? Select two options. She should have used division to represent the quotient of six and x. She should not have included an equal sign. She should have used parentheses to show that Negative 2 is multiplied by (x minus 4.5). She should have subtracted the difference (x minus 4.5) from Negative 2. She should have used parentheses to show that Negative 2 is multiplied by (4.5 minus x).
Answers
The errors Emma made are
She should have used division to represent the quotient of six and xShe should have used parentheses to show that - 2 is multiplied by (x - 4.5)Writing an expression as an equationFrom the question, we are to determine the errors that Emma made
The given statement is
"The quotient of six and a number, x, is the same as negative two times the difference of x and 4.5"
"The quotient of six and a number, x", can be expressed as
6/x
"Two times the difference of x and 4.5", can be expressed as
2(x - 4.5)
Hence, the errors Emma made are
She should have used division to represent the quotient of six and xShe should have used parentheses to show that - 2 is multiplied by (x - 4.5)Learn more on Writing an expression as an equation here: https://brainly.com/question/13155862
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Consider triangle ABC with the measure of an angle B = 60ﾟ and sidelinks a equal 4 &c equal 5 what option listed is an expression that is equivalent to the length of side B
Answers
The law of cosine helps us to know the third side of a triangle when two sides of the triangle are already known the angle opposite to the third side is given. The correct option is B.
What is the Law of Cosine?The law of cosine helps us to know the third side of a triangle when two sides of the triangle are already known the angle opposite to the third side is given. It is given by the formula,
[tex]c =\sqrt{a^2 + b^2 -2ab\cdot Cos\theta}[/tex]
where
c is the third side of the triangle
a and b are the other two sides of the triangle,
and θ is the angle opposite to the third side, therefore, opposite to side c.
The length of the sidelink b using the cosine rule can be written as,
[tex]b = \sqrt{a^2+c^2-2ac\cos(\angle B)}\\\\b = \sqrt{4^2+5^2 - 2(4)(5)(\cos 60^o)}\\\\b = \sqrt{16+25-20}[/tex]
Hence, the correct option is B.
The complete question is:
Consider ABC with the measure of angle B equal to 60 degrees, and side lengths a=4 and c=5. Which option lists an expression that is equivalent to the length of side b?
Options are given in the image below.
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Which of the following is the most profitable investment for a candy shop that earns $1 revenue per pound of candy? (5 points) Group of answer choices Worker at $10 per hour, producing eight pounds of candy per hour Worker at $12 per hour, producing 16 pounds of candy per hour Machine with $5 per hour operating cost, producing 10 pounds of candy per hour Machine with $8 per hour operating cost, producing 14 pounds of candy per hour
Answers
Then the most profitable investment is the first machine (third option) because for each dollar, it produces 2 pounds of candy.
Which of the following is the most profitable investment?The first option is a worker that costs $10 per hour and produces 8lb, then this worker gives:
8lb/$10= 0.8 pounds per dollar.
The second worker costs $12 per hour, and produces 16lb per hour, so this worker gives:
16lb/$12 = 1.33 pounds per dollar.
The third option is a machine that cost $5 per hour and produces 10lb per hour, then:
10lb/$5 = 2 pound per dollar.
The last option costs $8 per hour, and produces 14 pounds per hour, then:
14lb/$8 = 1.75 pounds per dollar.
Then the most profitable investment is the first machine (third option) because for each dollar, it produces 2 pounds of candy.
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Please I need help with this 2 calc questionssss. Thank you
Answers
Question 14
As we approaching from the negative side, [tex]|x-3|=3-x[/tex].
So, we have [tex]\lim_{x \to 3^{-}}=\frac{3-x}{x-3}=\frac{-(x-3)}{x-3}=\boxed{-1}[/tex]
and thus the limit exists at x=3.
Question 15
For the top part of the function, [tex](9-9)^{2}=0[/tex]
For the bottom part of the function, [tex]9-9=0[/tex]
As both one-sided limits are equal, L = 0, and thus the limit exists at x=9.
A chocolate bar measures 40 mm wide, 80 mm long, and 5 and 1 over 2 mm high. Will’s Wrapping Company is making a wrapper to cover the chocolate bar. Use the correct net and determine how much paper will be needed to make the wrapper.
Answers
Answer:
7720 mm²
Step-by-step explanation:
area = 2 × 80 mm × 40 mm + 2 × 80 mm × 5.5 mm + 2 × 40 mm × 5.5 mm
area = 7720 mm²
if anyone knows how to answer this do so please
Answers
Answer:
b = 5.3
A = 48.6
B = 41.4
Step-by-step explanation:
Attached is the Pics contain the Pythagoras Theorem and Trigonometric Identities.
So for our Right-Angled Triangle we have from Pythagoras Identity that
[tex]a^{2} +b^{2} =c^{2}[/tex] ,, Then [tex]b^{2} =c^{2} -a^{2}[/tex] where a=6 , c=8
by substituting,
[tex]b^2=8^2-6^2=28\\ b=\sqrt{28} =2\sqrt{7}=5.3[/tex]
Now to find A , we will use the Trigonometric Identities
Since [tex]sin(A)=\frac{Opposite}{Hypotenuse}=\frac{a}{c} =\frac{6}{8}=\frac{3}{4} \\[/tex]
[tex]A=sin^{-1} (\frac{3}{4} )=48.6\\[/tex]
Since in any Triangle A+B+C=180
Then B=180-A-C=180-48.6-90=41.4
Find the slope of the line that contains the points:
(-4, 3) and (4, 9)
a) 3/4
b) -3/4
c) 4/3
d) -4/3
Answers
Step-by-step explanation:
ATTACHED IS THE SOLUTION!!!Answer:
The answer would be A. 3/4
Step-by-step explanation:
To find the slope, you divide the difference of the y-coordinates of 2 points on a line by the difference of the x-coordinates of those same 2 points.
Huilans age is three times Thomas age The sum of their ages is 76 what is Thomas age
Answers
Answer:
let x be thomas age
3x for huilans age since huilans age is 3 time thomas age
both of their ages add to 76
ie. x + 3x =76
solve for x
4x = 76
x = 76 ÷ 4
x = 19. thomas age
to get huilans age : take 19 × 3 = 57
If an object moves along the y-axis (marked in feet) so that it’s position at time X (in seconds) is given by f(x) = 108x -12x^2 , find the following:
Answers
The velocity equation is:
[tex]v(x) = 108 - 24x[/tex]
Such that:
[tex]v(0) = 108 ft/s\\v(2) = 60ft/s\\[/tex]
And the velocity is zero for x = 4.5
How to find the velocity equation?
We know that the position equation is:
[tex]f(x) = 108*x - 12*x^2[/tex]
To get the velocity, we need to differentiate with respect to x, so we get:
[tex]v(x) = \frac{df(x)}{dx} = 108 - 2*12*x = 108 - 24*x[/tex]
Now we want to get the velocity for x = 0 and x = 2, we will get:
[tex]v(0) = 108 - 24*0 = 108\\\\v(2) = 108 - 24*2 = 60[/tex]
Then, when x = 0, the velocity is 108 ft/s, when x = 2 the velocity is 60 ft/2.
Finally, we need to solve:
[tex]v(x) = 108 - 24*x = 0\\\\x = 108/24 = 4.5[/tex]
The velocity is zero when x = 4.5
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A home cleaning service takes more time to clean larger spaces. The table shows the number of minutes it takes for the service to clean a certain number of square feet within a home. Square Feet and Cleaning Time Number of Square Feet, x 50 80 100 150 Time in Minutes, y 20 32 40 60 Based on the table, how many minutes would it take for the service to clean 200 square feet? 20 minutes 80 minutes 125 minutes 400 minutes
Answers
The time required to clean 200 square feet will be 80 minutes. Option B is correct.
What is the surface area?The total land area of all the faces of a three-dimensional object is its surface area. When we wish to wrap something in real life, we employ the notion of surface areas of distinct things.
Let x be the cleaning time for the 200 square feet area;
From the table, it is observed that for a 1 sq. foot area the time needed to clean will be 0.4 minutes.
1 square feet = 0.4 minute
200 square feet = 200 × 0.4
200 square feet = 80 minute
The time required to clean 200 square feet will be 80 minutes.
Hence, option B is correct.
The table is attached in the attachment.
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When you brush your wig will all the hair come off the wig completely or will it still be the same in a year
Answers
Answer:
no
Step-by-step explanation:
I need help on these 2 questions please ASAP
Answers
Step-by-step explanation:
Q9
a) :
First, find the common difference for the sequence. add the first term to the second term
3+3= 6
add second term to the third term
3+6=9
add third term to the fourth term
6+9=15
b) 15 +24= 39
24+39=63
39+63= 102
Q10
a) 2 , 10 , 8
b) find the common difference for the sequence. Subtract the first term from the second term.
10-2= 8
Subtract the second term from the third term
8-10= -2
c) 10
Pls help! I need to finish quickly.
Answers
The inverse of the function f(x) = 2x + 5 is; f⁻¹(x) = (x - 5)/2
How to find the Inverse of a Function?An inverse function in mathematics is function which reverses the another function.
We want to find the inverse of the function; f(x) = 2x + 5
Thus;
Step 1; y = 2x + 5
Step 2; Subtract 5 from both sides to get;
y - 5 = 2x
Step 3; Since we want to find the inverse of f(x), then we now have;
x - 5 = 2y
Step 4: Divide both sides by 2 to get;
(x - 5)/2 = y
Step 5; Thus, we have;
f⁻¹(x) = (x - 5)/2
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Write the equation 2x + 4y = 12 in slope intercept form. Show your work.
Answers
Answer:
First we have to find the value of x;
then we have to find the value of y;
Step-by-step explanation:
here,
2x= 12
x = 12\2
x = 6
now,
4y = 12
y = 12\4
y = 3
° the value of x is 6
° ° the value of y is 3
Four dragons (red, yellow, green, and blue) are getting ready to perform. The red dragon and the yellow dragon do not want to go on stage first. In how many different ways can the dragons go on stage?
Answers
Using the arrangements formula, it is found that there are 12 ways for the dragons to go on stage.
What is the arrangements formula?The number of possible arrangements of n elements is given by the factorial of n, that is:
[tex]A_n = n![/tex]
In this problem, we have that:
For the first dragon, either the green or the blue dragon, hence two outcomes.For the remaining dragons, there are 3! possible ways.Hence the number of ways by which they can go on stage is:
N = 2 x 3! = 12.
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Answer:
Using the arrangements formula, it is found that there are 12 ways for the dragons to go on stage.
What is the arrangements formula?
The number of possible arrangements of n elements is given by the factorial of n, that is:
A_n = n!A
n
=n!
In this problem, we have that:
For the first dragon, either the green or the blue dragon, hence two outcomes.
For the remaining dragons, there are 3! possible ways.
Hence the number of ways by which they can go on stage is:
N = 2 x 3! = 12.
Select the procedure that can be used to show the converse of the Pythagorean theorem using side lengths chosen from 3 feet, 4 feet, 5 feet, and 6 feet.
A. Knowing that 32 + 42 = 52, draw the 3-foot side and the 4-foot side with a right angle between them. The 5-foot side will fit to form a right triangle.
B. Knowing that 32 + 52 ≠ 62, draw the 3-foot side and the 5-foot side with a right angle between them. The 6-foot side will fit to form a right triangle.
C. Knowing that 32 + 42 = 52, draw any two of the sides with a right angle between them. The third side will fit to form a right triangle.
D. Knowing that 42 + 52 < 62, draw the 4-foot side and the 5-foot side with a right angle between them. The 6-foot side will fit to form a right triangle.
Answers
Using Pythagoras theorem, we know that 3² + 4² = 5², draw the 3-foot side and the 4-foot side with a right angle between them. The 5-foot side will fit to form a right triangle
How to use Pythagoras theorem to prove a right triangle?The Pythagoras theorem states that the sum of two squares equals the squared of the longest side.
Therefore,
c²= a²+ b²
where
c = hypotenuse sidea and b are the other legs.Therefore,
3² + 4² = 5²
Hence,
Knowing that 3² + 4² = 5², draw the 3-foot side and the 4-foot side with a right angle between them. The 5-foot side will fit to form a right triangle
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I have redone this one assignment like 5 times and I keep getting it wrong help
Answers
5/9 * 10/10 = 50/90
77/90 which is about 3/4 mile!
(05.05 HC) Figure ABCD has vertices A(-3, 2), B(2, 2), C(2, -4), and D(-3, -2). What is the area of Figure ABCD?
Answers
Answer:
I don't know how to answer this, sorry!
Step-by-step explanation:
.
do synethtic wig curls only come out if you put heat on it Answer Asap for brainlist
Yes or no
Answers
Answer:
There is no definitive answer to this question since different wigs are made with different kinds of synthetic fibers. Some synthetic fibers are more resistant to heat than others, so it really depends on the specific wig. In general, however, it is generally not recommended to use heat on synthetic wigs, as it can damage the fibers.
Solve inequality
58 - 10xs 20 + 9x.
Answers
Answer:
x>=2 so B
Hope this helps :)
Keisha borrowed money from a bank to buy a fishing boat.
She took out a personal, amortized loan for $19,500, at an interest rate of 4.75%, with monthly payments for a term of 5 years.
(a) Find Keisha's monthly payment.
(b) If Keisha pays the monthly payment each month for the full term,
find her total amount to repay the loan.
$0
(c) If Keisha pays the monthly payment each month for the full term,
find the total amount of interest she will pay.
$0
Answers
Answer:
Your $19,513 loan has 60 monthly payments of $366.00 at 4.75%.
If you make all of your payments on this loan, and do not prepay any of the principal, the total interest for this loan is $2,447.26.
Step-by-step explanation:Loan Summary
Total payments $21,960.26
Total interest paid $2,447.26
Loan amount $19,513
Monthly payment $366.00
Interest rate 4.75%
Term 60 monthly payments
Use the trigonometric ratios sine, cosine, and tangent to answer the following questions
Answers
Answer:
Step-by-step explanation:
A gas can hold 10 litters of gas.
How many could we fill 7 litters of gas?
Answers
Answer:
a gas can hold 7×10=70 litters of gas
+
120
A. Right
B. Obtuse
C. Acute
D. Equilateral
E. Scalene
OF. Isosceles
Answers
Answer:
See below
Step-by-step explanation:
Isosceles because two sides are same length
and
Obtuse because there is an interior angle greater than 90 °
The junior class sold 75% of the raffle tickets. They started with 100 raffle tickets.
How many tickets have been sold?
Answers
Answer:
75
Step-by-step explanation:
Since they sold 75% of the raffle tickets and the total is 100 raffle tickets, you simply need to find 75% of 100. To find x% of a, you simply convert x% to a decimal by dividing by 100 and then multiplying it by a. So the general formula to find x% of a would be: [tex]a*\frac{x}{100}[/tex]. In this case a=100, and x=75. So you have the equation: [tex]100*\frac{75}{100} = \frac{100 * 75}{100}=\frac{75}{1} = 75[/tex]
Answer: 75 tickets
Step-by-step explanation:
- percentages are out of 100, they are a rate, number, or amount in each hundred
- for example, (10/100) x 100 (because it’s out of 100) = 10%
- or it could be (6/40) x 100 (because it’s out of 100) = 15%
- so because they started with 100 tickets, it’s just (75/100) x 100 =75
hope this helps :)