Answers
Step-by-step explanation:
So generally whenever you have a problem like this, you want to factor out the denominator and numerator, that way when dividing, you can easily see what the domain is by looking at factors.
So let's start by factoring the numerator and denominator of f(x)
[tex]f(x) = \frac{3x-6}{2x+10}\implies \frac{3(x-2)}{2(x+5)}[/tex]
Now let's do this with the g(x)
[tex]g(x) = \frac{-2x+8}{x+2}\implies\frac{-2(x-4)}{x+2}[/tex]
Now when you divide f(x) by g(x), you keep f(x), change division to multiplication, and flip the g(x) so it's the reciprocal. This gives you the expression
[tex]\frac{f(x)}{g(x)}=\frac{3(x-2)}{2(x+5)}*\frac{x+2}{-2(x-4)}[/tex]
Now just multiply the numerator and denominator
[tex]\frac{f(x)}{g(x)}=\frac{3(x-2)(x+2)}{2(x+5)*-2(x-4)}[/tex]
So the main point of factoring was to easily see when the denominator is zero, (since if one of the factors is zero, the entire thing will be zero, since zero * anything = zero), but it was also to find any removable discontinuities. Which are just when you have factor in the denominator and numerator, so you can simplify the fraction further by canceling it out. To give an example: [tex]f(x)=\frac{(x-2)(x+4)}{(x-2)(x+3)}\implies\frac{x+4}{x+3}[/tex]. But the domain of f(x) is still restricted so that: [tex]x\ne 2, -3[/tex]. But there will not be a vertical asymptote at x=2 like with traditional domain restrictions in a rational function. This is because you can technically define f(2), but only if you use the most simplified form. This will result in a hole being at that point to signify that it's a removable discontinuity.
Anyways a bit off topic, but something that may be necessary in other examples. Anyways, now to find the domain, we just need to find when f(x)/g(x) is not defined. Or more specifically we find when the denominator is going to be zero. So we just need to set both factors equal to zero.
[tex]0 = 2(x+5)\\\text{divide both sides by 2 (a bit redundant)}\\0 = x+5\\\text{subtract 5 from both sides}\\-5 = x[/tex]
[tex]0 = -2(x-4)\\\text{divide both sides by -2 (a bit redundant)}\\0 = x-4\\\text{add 4 to both sides}\\x=4[/tex]
So the domain restrictions would be: [tex]x\ne-5, 4[/tex]
This can be expressed in interval notation: [tex](-\infty, -5) \cup (-5, 4) \cup (4, \infty)[/tex].
You can also express this in terms of set builder notation: [tex]D: \{x|x\ne-5, x\ne4\}[/tex]
Related Questions
Is the set of the following numbers ordered from greatest to least or least to greatest
Answers
Answer: Your answer should be least to greatest I believe
Using the Distributive Property, rewrite 2x + 16 as a product of two factors. *
1. 2x + 8
2. 2(16x)
3. 2(x + 8)
4. 2(x + 16)
Answers
if you distribute the 2 into the parenthesis, 2(x) + 2(8), the result will be 2x+16
You enter an elevator and go down 7 floors you exit on the 10th floor. Write an equation you can use to find the floor where you entered the elevator
Answers
(0 - 7) - 3
Name all digits that can be the last digit of prime numbers.
Answers
Answer:
1,2,3,5,7,9
Step-by-step explanation:
I started by seeing all the digits (0-9) which are prime numbers.
(05.07 MC)
The net of a square pyramid is shown below
3 inches
2 inches
3 inches
2 inches
What is the surface area of the solid?
16 square inches
24 square inches
28 square inches
Previous Question
Question 1 (Answered)
Answers
Answer:
16 2x3 divided by 2 x4=12 + 2x2=16 I think hopefully this helps :)
This is the question I need the answer in 24 house or less please and thank u
Answers
Answer:
Vertex: (–9, 1)
Axis of Symmetry: x = –9
Step-by-step explanation:
x = 9 is the lowest point on the graph. This makes it the x-coordinate of the vertex and the axis of symmetry. Now count how many squares above the x-axis it is. It's 1 square above, right? This means the vertex is (–9, 1).
Hope this helps you. Have a great day!
The first term of an Ap are x, 2x +1, and 5x+1 Find
x and the sum of the
l0 terms
first
Answers
Answer:
145/2
Step-by-step explanation:
d=[2x+1-x] = [5x+1 -(2x+1)]
x+1=3x
x-3x=-1
-2x=-1
x=1/2
therefore first term is 1/2
second term is 2(1/2)+1= 2
common difference d is 2-1/2=3/2
sum of first n terms is n/2 [2a+(n-1)d] where n=10
sum of first 10 terms= 10/2[2×1/2+(10-1)3/2]
=5[1+(9)3/2] =5[1+27/2] = 5[29/2] =145/2
A submarine is 84 feet below the surface of the water and descends 10 feet
deeper every minute. How many minutes will it take for the submarine to be
located 219 feet below the surface? Write and solve an equation,
Answers
Answer:
13.5
Step-by-step explanation:
219-84=135
10*13.5=135
so it would take 13.5 minutes to get to 219 since 84+135=219
Jocelyn boils 24 eggs. From experience, she knows that the shell of any given during the boiling process. If shells crack independently, what is the probability the boiling process with a cracked shell? Round to 2 decimal places.
Answers
Answer:
.62
Step-by-step explanation:
Keep it nice and simple.
Answer:
.62
Step-by-step explanation:
The solution set in interval notation is ??
Answers
Answer:
Step-by-step explanation:
Factor out the GCF from the terms of the polynomial 2x^3 − 5x^2 + 25.
Answers
Answer:
Step-by-step explanation:
The polynomial is already factored.
A playground is 45 feet wide
convert the wind to yards
Answers
P-250 ft
Find the dimensions of the rectangle
The perimeter is given.
W
3w- 15
Answers
Answer:
A = L × W A = L × W A=L×W.
A = L × W A = L × W A=L×W.
P = 2 L + 2 W P = 2L + 2W P=2L+2W.
show that x²/√x²+4 is continuous at x=2
Answers
Answer:
[tex]\lim_{x \to 2} \frac{x^{2} }{x^{2} +4} = \frac{1}{2} = f(C)[/tex]
The function f(x) is continuous
Step-by-step explanation:
Explanation:-
Given that the function
[tex]f(x) = \frac{x^{2} }{x^{2} +4}[/tex]
put x = c =2
[tex]f(c) = f(2) = \frac{(2)^{2} }{(2)^{2} +4} = \frac{4}{8} = \frac{1}{2}[/tex]
[tex]\lim_{x \to 2} \frac{x^{2} }{x^{2} +4} = \frac{2^{2} }{2^{2} +4} = \frac{4}{4+4} = \frac{4}{8}[/tex]
[tex]\lim_{x \to 2} \frac{x^{2} }{x^{2} +4} = \frac{1}{2}[/tex]
[tex]\lim_{x \to 2} \frac{x^{2} }{x^{2} +4} = \frac{1}{2} = f(C)[/tex]
Given function f(x) is continuous
The earth travels 1,050 miles per hour.
At this rate, how many miles does the earth travel per day?
Plz Help God will Bless You!!
Answers
The earth travels 1,050 miles in an hour.
There are 24 hours in a day.
So multiply 1,050 by 24.
And you get 25,200.
The earth travels 25,200 miles in a day.
Un jugador encesta con probabilidad de 0.73 Calcula la probabilidad de que al tirar 6 veces enceste:
a) al menos 4 veces
b) ninguna vez
c)todas las beces
d)dos veces
e) menos de tres veces
f) 5 o mas veces
Answers
do not count the
individual
oranges.
count them as
groups.
Answers
Answer:
A. 5
B. 100
C. 300
PLEASE HELP !! ILL GIVE BRAINLIEST *EXTRA POINTS*..
IM GIVING 40 POINTS !! DONT SKIP :((.
Answers
Negative nine is greater than or equal to the sum of two - fifths times a number and 9.
Answers
Answer:
W is either -22.5, or less than -22.5
Step-by-step explanation:
-9>=2/5 times w (I wasn't sure what you meant by "and" 9. Like 9w?)
W has to be a negative number, or else it would be greater than 9, which would make the equation wrong.
2/5 times -22 1/2=-9. So w can be -22 1/2 or -22.5
W can then be anything "lower" (on the left side) of -22.5.
W<-22.5
twice the square root of a number is 12
Answers
Let us assume that the number is X
According to the statement:
2✓x = 12
Divide both sides by 2;
✓2 = 6
Square both sides,
x² = 6²
X = 36
Hence the required number is 36.
8) Which of the Prime number are
even numbers?
C
a) 0 & 2
b) 1 & 2
c) 2
ОО
d) None
Answers
how many odd _2-digit positive integers can be written using the numbers 2,4,5, and 8?
Answers
Answer:
4 different odd integers: 25, 45, 55, 85
Step-by-step explanation:
Do
10 cm
10 cm
20 cm
form an triangle?
Answers
Answer:
No it is unable to form a triangle.
Step-by-step explanation:
It cannot be because one side is two times the other side, and there are two of the "other sides". It cannot form a triangle.
Answer:
Step-by-step explanation:
A committee of 3 is to be formed from 5 men and 6 women. (a) What is the probability that a random committee is made up of 2 women and 1 man? (b) Suppose now that the committee consists of different positions: Chairperson, Vice-Chairperson, and Treasurer. What is the probability that the Chairperson and Vice Chairperson are both women and the Treasurer is a man?
Answers
Answer:
a) 0.4545 = 45.45% probability that a random committee is made up of 2 women and 1 man.
b) 0.1515 = 15.15% probability that the Chairperson and Vice Chairperson are both women and the Treasurer is a man
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
If the order of the people chosen is important, we use the permutations formula. If the order is not important, we use the combinations formula.
Permutations formula:
The number of possible permutations of x elements from a set of n elements is given by the following formula:
[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
(a) What is the probability that a random committee is made up of 2 women and 1 man?
Order is not important, so we use the combinations formula.
Desired outcomes:
2 women from a set of 6, and a men from a set of 5. So
[tex]D = C_{6,2}*C_{5,1} = \frac{6!}{2!4!}*\frac{5!}{1!4!} = 15*5 = 75[/tex]
Total outcomes:
3 people from a set of 6 + 5 = 11. So
[tex]T = C_{11,3} = \frac{11!}{3!8!} = 165[/tex]
Probability:
[tex]p = \frac{D}{T} = \frac{75}{165} = 0.4545[/tex]
0.4545 = 45.45% probability that a random committee is made up of 2 women and 1 man.
(b) Suppose now that the committee consists of different positions: Chairperson, Vice-Chairperson, and Treasurer. What is the probability that the Chairperson and Vice Chairperson are both women and the Treasurer is a man?
Now, the order is important, which means that we use the permutations formula.
Desired outcomes:
2 women from a set of 6, and a men from a set of 5. So
[tex]D = P_{6,2}*P_{5,1} = \frac{6!}{4!}*\frac{5!}{4!} = 30*5 = 150[/tex]
Total outcomes:
3 people from a set of 11. So
[tex]T = P_{11,3} = \frac{11!}{8!} = 990[/tex]
Probability:
[tex]p = \frac{D}{T} = \frac{150}{990} = 0.1515[/tex]
0.1515 = 15.15% probability that the Chairperson and Vice Chairperson are both women and the Treasurer is a man
A bucket contains 13 & 1/3 gallons of water and is 5/6 full. How many gallons of water would be in a full bucket?
Answers
Answer:
16
Step-by-step explanation:
Answer:
17 2+2+2+2+2+2+2+2+1
Step-by-step explanation:
the bucket contains 13 and 6
Can someone explain to me how to do this problem?
x • 1 + x/1 =
Answers
x + x/1
2x/1
=2x
find the perimeter and area of the rectangle with vertices (1, -2), (1, 7), (-3, 7), and (-3, -2)
Answers
Answer:
26
Step-by-step explanation:
9+4+9+4=26
Here again are the results of a survey that students conducted at a mall. The students conducted this survey as part of a statistics project to determine if younger adults are more likely to have tattoos. At least one tattoo No tattoo Row Totals Age 18 - 29 170 320 490 Age 30 - 50 60 445 505 Column Totals 230 765 995 Which proportions are most useful in analyzing the relationship between age and tattoos? Group of answer choices 170 / 230 and 60 / 230
Answers
Answer:
1. 170/490 and 60/505
2. 320/490 and 445/505
Step-by-step explanation:
No options were given, but the question can still be answered
Given
[tex]\begin{array}{cccc}{} & {\ge 1\ tattoo} & {0\ tattoo} & {Row\ Total} & {18 -29} & {170} & {320} & {490} & {30 - 50} & {60} & {445} & {505} & {Column\ Total} & {230} & {765} & {995}\ \end{array}[/tex]
Required
Which proportion best compares age and tattoos?
The row data represents the age range of each group while the column data represents the amount of tattoo by each group.
So, to make comparison:
You simply pick a cell data and compare it to the row total.
Take for instance
For those who have at least 1 tattoo. The proportion to use are:
170/490 --- For age range: 18 - 29
60/505 --- For age range: 30 - 29
For those without tattoo. The proportion to use are:
320/490 --- For age range: 18 - 29
445/505 --- For age range: 30 - 29
5,3,1,-1, common difference or common ratio
Answers
Answer:
Common
Step-by-step explanation:
Probability is a measure of how likely an event is to occur. Choose the probability that best matches each of the following statements: (a) This event is impossible: [ Select ] (b) This event will occur more often than not, but is not extremely likely: [ Select ] (c) This event is extremely unlikely, but it will occur once in a while in a long sequence of trials: [ Select ] (d) This event will occur for sure: [ Select ]
Answers
Answer:
(a) 0
(b) 0.6
(c) 0.01
(d) 1
Step-by-step explanation:
Given
Probabilities: 0, 0.01, 0.3, 0.6, 0.99, and 1 ---> Missing information
Required
Match the probabilities with appropriate values
(a) Impossible:
An impossible event will always have a probability of 0
(b) Occurrence is more often than not, but not extremely likely:
An event that will occur more often than not will have a probability greater than 0.5 but less than 1
However, if the likelihood is not extremely likely, the probability will be closer to 0.5 and 1.
The probability that matches this description is 0.6
(c) Occurrence is extremely unlikely, but will occur once in a while
An event that is extremely unlikely will have a value greater than 0 and the value will be very close to 0.
The probability that matches this description is 0.01
(d) A certain event:
A certain event has a probability of 1