Answer:
Statement b is true; statement a is false
Step-by-step explanation:
Best to write out (f o g)(x) and then determine its domain:
"(f o g)(x)" indicates that g(x) is used as the input to f(x):
(f o g)(x) = 5(√x+6) + 7, or 5√x + 30 + 7, or 5√x + 37.
The domain of this composite function is [0, infinity).
Thus, statement b is true: "9 is part of the domain of (f o g)(x) = 5(√x+6) + 7"
To determine which statement is true, we will need to look at the functions f(x) and g(x) and their composition (f ° g)(x), which means we plug g(x) into f(x).
The given functions are:
f(x) = 5x + 7
g(x) = √x + 6
The composition of the functions (f ° g)(x) is f applied to g(x), which would be:
(f ° g)(x) = f(g(x)) = f(√x + 6)
This simplifies to:
(f ° g)(x) = 5(√x + 6) + 7
To be able to plug a number into (f ° g)(x), it must first be in the domain of g(x), and then the result of g(x) must be in the domain of f(x). The domain of g(x) is defined by the set of all x for which g(x) is real and defined. Since g(x) includes a square root, x must be non-negative (x ≥ 0). For f(x), the domain is all real numbers since linear functions are defined for all real x.
So, to determine if 9 is in the domain of (f ° g)(x), we will do the following:
1. Verify if 9 is in the domain of g(x). Since the domain of g(x) includes all x ≥ 0 and 9 ≥ 0, 9 is in the domain of g(x).
2. Calculate g(9) to determine if the output is within the domain of f(x):
g(9) = √9 + 6
g(9) = 3 + 6 (because √9 is 3)
g(9) = 9, which is a real number and thus in the domain of f(x).
3. Since we have that g(9) is in the domain of f(x), we can compute (f ° g)(9):
(f ° g)(9) = f(g(9))
(f ° g)(9) = f(9)
(f ° g)(9) = 5 * 9 + 7
(f ° g)(9) = 45 + 7
(f ° g)(9) = 52
Thus, we have found that 9 is in the domain of g(x), and that (f ° g)(9) is defined and equals 52. This means that statement B, "9 is in the domain of f ° g," is true.
Which graph represents the compound inequality? n<-2 or n 24
3 is correct, because for n more than 4 we must have node on x=4 too
Need help fast do not understand this one. Solve: (x+5) / (x+8)=1+(6) / (x+1) showing all work.
Answer: [tex]x=-\frac{17}{3}[/tex]
Step-by-step explanation:
Given the equation [tex]\frac{(x+5)}{(x+8)}=1+\frac{6}{(x+1)}[/tex], you need to make the addtition indicated on the right side of the equation:
[tex]\frac{(x+5)}{(x+8)}=\frac{(x+1)+6}{(x+1)}\\\\\frac{(x+5)}{(x+8)}=\frac{(x+7)}{(x+1)}[/tex]
Now, multiply both sides of the equation by (x+8) and (x+1):
[tex](x+1)(x+8)\frac{(x+5)}{(x+8)}=\frac{(x+7)}{(x+1)}(x+1)(x+8)\\\\(x+1)(x+5)=(x+7)(x+8)[/tex]
Now, apply Distributive property:
[tex]x^2+5x+x+5=x^2+8x+7x+56[/tex]
Simplifying, you get:
[tex]6x+5=15x+56[/tex]
Subtract 5 and 15x from both sides:
[tex]6x+5-15x-5=15x+56-15x-5\\\\-9x=51[/tex]
Finally, divide both sidesby -9:
[tex]\frac{-9x}{-9}=\frac{51}{-9}\\\\x=-\frac{17}{3}[/tex]
the factored form of a quadratic equation is x (x-4). the ordered pair (0,0) represents one of the zeros of the associated quadratic function. which ordered pair represents the other zero?
Answer:
the other zero is represented by the ordered pair (4, 0)
Step-by-step explanation:
We have the factored expression
[tex]x (x-4)[/tex]
We must find out for what values of x this function is equal to zero.
Then we equal the function to zero and solve for the variable x
[tex]x (x-4)=0[/tex]
The function is equal to zero when one of the two terms of the product is equal to zero.
So
[tex]x =0[/tex] We already know that this is a solution
or
[tex]x-4=0\\x =4[/tex]
Then the other zero is represented by the ordered pair (4, 0)
Determine which diagram could be used to prove ABC - EDC using similarity transformations.
Answer: A The first Answer to the left
Step-by-step explanation:
The diagram could be used to prove that ABC-EDC using similarity transformations is option A.
The first Answer to the left
We have given that,
The diagram could be used to prove ABC - EDC using similarity transformations.
What is the transformation?
A similarity transformation is one or more rigid transformations (reflection, rotation, translation) followed by a dilation.
We have to determine which diagram could be used to prove ABC-EDC.
Therefore the answer is the first diagram.
The diagram could be used to prove that ABC-EDC using similarity transformations is option A.
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Mike spent $7920 on his vacation, which was 11% of his monthly salary. What was his monthly salary?
Answer : 72,000
Ex: Let x be his monthly salary.
The equation to set up would be
7920=0.11x
This equation represents 11% of monthly salary = 7920
To solve you divide 0.11x by 0.11 to get x by itself. then you do what you do to one side to the other. therefore you divide 7920 by 0.11 and you get 72,000
What is the slope of this line?
Answer: [tex]-\frac{8}{5}[/tex]
Remember: RISE/RUN (y/x). Lines that are increasing have a positive slope, and lines that are decreasing have a negative slope.
You can find the slope in two ways:
1. Useful if the line is graphed: count the units between 2 points on the line.
Let's use the points (-1, 4) and (4, -4).(-1, 4) is 8 units higher than (4, -4) and 5 units to the left of (4, -4).Because the line is decreasing, the slope is negative.Therefore, the slope is [tex]-\frac{8}{5}[/tex].2. Useful if the line is not graphed: find the difference between the y-coordinate values divided by the difference of the x-coordinate values.
Let's use the points (-1, 4) and (4, -4).[tex]\frac{-4 - 4}{4 - (-1)} = \frac{-8}{5}[/tex]Therefore, the slope is [tex]-\frac{8}{5}[/tex].Answer:
[tex]m=-\frac{8}{5}[/tex]
Step-by-step explanation:
Let
[tex]A(-1,4),B(4,-4)[/tex]
we know that
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
substitute the values
[tex]m=\frac{-4-4}{4+1}[/tex]
[tex]m=\frac{-8}{5}[/tex]
[tex]m=-\frac{8}{5}[/tex]
Compare the monthly payment amount of Annabelle's dream car at two different car dealerships.
Dealership A: The car costs $30,000, and the loan has an annual interest rate of 4.8%.
Dealership B: The car costs $29,800, and the loan has an annual interest rate of 5.4%.
Determine the monthly payment for each dealership, and decide which is cheaper. Both interest rates are compounded
monthly. Both loans are for 5 years, or 60 months. Assume that there is no down payment.
Answer:
Dealership A is cheaper. Hope it helps.
Answer:
By comparing both the payments we can say that Dealer A is cheaper by $4.50.
Step-by-step explanation:
Dealership A: The car costs $30,000, and the loan has an annual interest rate of 4.8% for 5 years.
The EMI formula is :
[tex]\frac{p\times r\times(1+r)^{n} }{(1+r)^{n}-1 }[/tex]
Now, p = 30000
r = [tex]4.8/12/100=0.004[/tex]
n = [tex]5\times12=60[/tex]
Putting values in formula we get;
[tex]\frac{30000\times0.004\times(1+0.004)^{60} }{(1+0.004)^{60}-1 }[/tex]
=> [tex]\frac{30000\times0.004\times(1.004)^{60} }{(1.004)^{60}-1 }[/tex]
EMI is = $563.34
Dealership B: The car costs $29,800, and the loan has an annual interest rate of 5.4% for 5 years.
p = 29800
r = [tex]5.4/12/100=0.0045[/tex]
n = [tex]5\times12=60[/tex]
Putting values in formula we get;
[tex]\frac{29800\times0.0045\times(1+0.0045)^{60} }{(1+0.0045)^{60}-1 }[/tex]
=> [tex]\frac{29800\times0.0045\times(1.0045)^{60} }{(1.0045)^{60}-1 }[/tex]
EMI = $567.84
By comparing both the payments we can say that Dealer A is cheaper by $4.50.
Find the value of each variable
Answer:
x=6.5
Step-by-step explanation:
This question is on intersecting cords and segments that intersect outside the circle
Forming an equation for this relation;
6(x+6)= 5(5+10)
6x+36=5(15)
6x+36=75.............................collect like terms
6x=75-36
6x=39...........................divide both sides by 6
x=6.5
9. The points (4, –6) and (9, -6) represent the location of two towns on a
coordinate grid, where one unit is equal to one mile. What is the distance, in miles, between the two towns?
Answer:
5 miles apart
Step-by-step explanation:
(4,-6) is 5 units away from (9,-6)
When it goes on sale for $1,600, Mikah buys a new plasma television that is valued at $1,800. He takes $300 from his savings account to use as a down payment and finances the rest
Mikah's assets increased by $1,500.
Answer: B
Step-by-step explanation:
Simplify 12a2b3 / 3ab
[tex] \frac{12 {a}^{2} {b}^{3} }{3ab } \\ = \frac{3 \times a \times b \times 4 \times a \times {b}^{2} }{3ab} \\ \\ cancelling \: common \: factors \\ \\ = 4a {b}^{2} [/tex]
Hence answer is 4ab²
Hope it helps...
Hope it helps...Regards;
Hope it helps...Regards;Leukonov/Olegion.
2. Tell which measure of central tendency best describes the data
3. Tell which measure of central tendency best describes the data
Answer:
Step-by-step explanation:
Mean, it is the average of all numbers and most accurately represents the data set.
The best measure of central tendency depends on the data set characteristics: the mean is suitable for outlier-free data, the median is better for skewed data or when outliers are present, and the mode indicates the most common value.
Explanation:Choosing the Best Measure of Central Tendency
When analyzing a data set, selecting the appropriate measure of central tendency is crucial. The three main measures are the mean, median, and mode. The mean is the arithmetic average and is best used when the data set does not contain outliers that can skew the results. The median is the middle value when the data is ordered and offers a better representation when outliers are present. The mode is the most frequently occurring value and provides insight into the most common data point.
In a symmetrical data set where the data are evenly distributed, all three measures of central tendency—the mean, median, and mode—will be equal or close to each other. In a skewed distribution or when the data contains outliers, the median is generally the most reliable measure of the center, as it is unaffected by extreme values.
Point C(3.6, -0.4) divides in the ratio 3 : 2. If the coordinates of A are (-6, 5), the coordinates of point B are .
If point D divides in the ratio 4 : 5, the coordinates of point D are .
Reset Next
Answer:
Point B is (10 , -4)
Point D is (10/9 , 1)
Step-by-step explanation:
* Lets revise the rule of the point which divides of a line segment in
a ratio
- If point (x , y) divides the line segment AB, where A is (x1 , y1) and
B is (x2 , y2) in the ratio m1 : m2
∴ x = [m2(x1) + m1(x2)]/(m1 + m2)
∴ y = [m2(y1) + m1(y2)]/(m1 + m2)
* Now lets solve the problem
- Point C (3.6 , -0.4) divides AB in the ratio 3 : 2, where A is (-6 , 5)
# x = 3.6 , y = -0.4
# A is (x1 , y1) , B is (x2 , y2)
∴ x1 = -6 , y1 = 5
∵ m1 : m2 = 3 : 2
- Substitute these values in the rule
∵ x = [m2(x1) + m1(x2)]/(m1 + m2)
∴ 3.6 = [2(-6) + 3(x2)]/(3 + 2)
∴ 3.6 = [-12 + 3x2]/5 ⇒ multiply both sides by 5
∴ 18 = -12 + 3x2 ⇒ add 12 to both sides
∴ 30 = 3x2 ⇒ divide both sides by 3
∴ 10 = x2
* The x-coordinate of B is 10
∵ y = [m2(y1) + m1(y2)]/(m1 + m2)
∴ -0.4 = [2(5) + 3(y2)]/(3 + 2)
∴ -0.4 = [10 + 3y]/5 ⇒ multiply both sides by 5
∴ -2 = 10 + 3y2 ⇒ subtract 10 from both sides
∴ -12 = 3x2 ⇒ divide both sides by 3
∴ -4 = y2
* The y-coordinate of B is -4
∴ Point B is (10 , -4)
- Point D divides AB in the ratio 4 : 5 where A (-6 , 5) and B (10 , -4)
- To find the coordinates of point D use the same rule above
# D is (x , y)
# A is (x1 , y1) and B is (x2 , y2)
# m1 : m2 is 4 : 5
∵ x1 = -6 and y1 = 5
∵ x2 = 10 and y2 = -4
∵ m1 = 4 and m2 = 5
- Substitute these values in the rule
∵ x = [m2(x1) + m1(x2)]/(m1 + m2)
∴ x = [5(-6) + 4(10)]/(4 + 5) ⇒ multiply the numbers
∴ x = [-30 + 40]/9 ⇒ add
∴ x = [10]/9 ⇒ Divide
∴ x = 10/9
* The x-coordinate of D is 10/9
∵ y = [m2(y1) + m1(y2)]/(m1 + m2)
∴ y = [5(5) + 4(-4)]/(5 + 4) ⇒ multiply the numbers
∴ y = [25 + -16]/9 ⇒ add
∴ y = [9]/9 ⇒ Divide
∴ y = 1
* The y-coordinate of point D is 1
∴ Point D is (10/9 , 1)
find the value of n such that b^2 + 16b + n is a perfect square trinomial
Answer:
64
Step-by-step explanation:
To complete the square
add ( half the coefficient of the b- term )² to b² + 16b
b² + 2(8)b + 8²
= b² + 16b + 64 = (b + 8)²
Someone help me with this question
Hello There!
The answer would be 0.6
This is because you have to multiply the probability of each of these. First, I know that 0.5 is 1/2 in fraction form and that 0.3 is 3/10
PLEASEE HEELPPP I NEED THIS QUESTION ANYONE HELP!!!
Answer:
4u= - 4i - 24j
u +w = -9i - 3j
Step-by-step explanation:
The question is on operation of vectors
Given u= -i -6j and w= -8i+3j
Then 4u = 4{ -i-6j}
4u= - 4i - 24j
u+w= -i -6j + -8i+3j......................collect like terms
= -i -8i + -6j +3j
= -9i + -3j
= -9i - 3j
Leila is considering buying her first home. The house she is interested in buying is priced at $125,000. Leila can put down a $20,000 payment, and she qualifies for a 30-year mortgage at 6%. What will her monthly mortgage payment be?
Answer:
* The monthly mortgage payment is $629.53 ⇒ answer C
Step-by-step explanation:
* Lets explain how to solve the problem
- Leila is considering buying her first home
- The house she is interested in buying is priced at $125,000
∴ She can put $20000 down payment
* Lets find the balance to be paid off on mortgage
∴ The balance = 125000 - 20000 = 105000
- She qualifies for a 30-year mortgage at 6%
* Lets find the rule of the monthly payment
∵ [tex]pmt=\frac{\frac{r}{n}[P(1+\frac{r}{n})^{tn}]}{(1+\frac{r}{n})^{tn}-1}[/tex] , where
- pmt is the monthly mortgage payment
- P = the initial amount
- r = the annual interest rate (decimal)
- n = the number of times that interest is compounded per unit t
- t = the time the money is invested or borrowed for
∵ P = 105000
∵ r = 6/100 = 0.06
∵ n = 12
∵ t = 30
∴ [tex]pmt=\frac{\frac{0.06}{12}[105000(1+\frac{0.06}{12})^{30(12)}}{(1+\frac{0.06}{12})^{30(12)}-1}[/tex]
∴ [tex]pmt=\frac{0.005[105000(1.005)^{360}]}{(1.005)^{360}-1} =629.528[/tex]
* The monthly mortgage payment is $629.53
Which of the following is an even function?
Answer:
I believe it is the third one (C)
Step-by-step explanation:
Hello There!
Your answer would be the first one.
f(x) = \x/
F of x has to equal f of -x
ASAP ANSWER
Bobby has 4 ½ gallons of soda to share. Each serving is 1 pint. How many servings does he have to share?
each gallon has 8 pints
4 [tex]\frac{1}{2}[/tex] * 8 = 36
so there would be 36 servings
Find the image of (–7, –3) reflected across the x-axis. A. (–7, 3) B. (7, –3) C. (7, 3) D. (–7, –3)
Answer:
(-7, -3) (Answer A)
Step-by-step explanation:
We start with the point (-7, -3). The x-coordinate does not change at all if we reflect this point across the x-axis. Whereas the y-value of (-7, -3) is -3, we end up with +3 after this reflection. The desired image is (-7, +3) (Answer A)
The coordinates of the point (-7, - 3) reflected along the x-axis is
(- 7, 3).
What are transformations?Two-dimensional figures can be transformed mathematically in order to travel about a plane or coordinate system.
Dilation: The preimage is scaled up or down to create the image.
Reflection: The picture is a preimage that has been reversed.
Rotation: Around a given point, the preimage is rotated to create the final image.
Translation: The image is translated and moved a fixed amount from the preimage.
We know reflection along the x-axis of a point (x, y) converts it into a
point (x, - y).
Given, A points (-7, - 3) is reflected along the x-axis.
Therefore, The new coordinates of the points (-7, - 3) reflected along the x-axis is (- 7, 3).
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Plz help me with this
Answer: C) the variance
Step-by-step explanation:
Variance is the "average" of the squared differences from the Mean.
Summer camp surveys 40 campus to see if they would take tennis next week. 12 camper say they would. If they were 250 campers. How many campers should the counselors plan on for next week’s tennis lesson
Answer:
76.8
Step-by-step explanation:
40/250 = 0.16 (1 campers vote)
0.16 x 12 = 1.92
1.92 x 40 = 76.8
To estimate the number of campers who might be interested in tennis lessons, a proportion can be used based on a survey of 40 campers, 12 of whom were interested. Scaling this up to the total 250 campers, approximately 75 campers might be interested in the tennis lessons.
The question is asking us to estimate the number of campers who would be interested in taking tennis lessons at a summer camp. From a small survey of 40 campers, 12 said they would take tennis lessons. To find out how many campers out of the total 250 might be interested, we need to use proportion. The proportion of campers interested in the survey is 12 out of 40, which we can set as equal to x out of 250, where x is the number we want to find. The equation is set up as follows:
12 / 40 = x / 250
Multiply both sides by 250 to isolate x:
250 × (12 / 40) = x
Calculate x:
x = 250 × (12 / 40) = 75
Therefore, the counselors should plan on approximately 75 campers attending next week’s tennis lesson.
kayla is sending invitations to 20 of her freinds for a pizza party. if there are 12 invitatons left to end out, how many invitations has she already mailed?
Kayla has mailed out 8 invitations
Use the pattern to answer the question. 27, 18, 12, 8. Which expression can be used to find the nth term of the pattern?
A. 27-6n
B. 27-6(n-1)
C. 27(2/3)^n
D. 27(2/3)^n-1
If you cross check with the answer , you’ll find option C to be the most appropriate option.
The difference between the first two terms is 9 and 9 is not a multiple of six, therefore first two are wrong.
Similarly, if you check option D for the first two terms , you get the second term as 27 which is false.
Therefore option C is the answer, also , it satisfies the series.
Final answer:
The correct expression to find the nth term of the given pattern is Option D: [tex]27(2/3)^{(n-1)}[/tex], as it accurately represents the geometric sequence of the provided numbers.
Explanation:
To find the nth term of the given pattern 27, 18, 12, 8, we will consider each option and see which one fits the sequence of numbers.
The first term (n=1) is 27.
The second term (n=2) is 18, which is 27 multiplied by ⅔ or 2/3.
The third term (n=3) is 12, which is the second term (18) multiplied by ⅔ or 2/3.
Similarly, the fourth term (n=4) is 8, which is the third term (12) multiplied by ⅔ or 2/3.
We can see that with each step, the term is being multiplied by a factor of ⅔ or 2/3. Therefore, an expression that uses multiplication by ⅔ repeatedly would likely represent the pattern.
Let's test the options:
Option A: 27-6n does not fit as it suggests a linear pattern with a constant subtraction, which does not match our sequence.
Option B: 27-6(n-1) also suggests a linear pattern and will not result in the given sequence when evaluated for n=1,2,3,4.
Option C: [tex]27(2/3)^n[/tex] represents a geometric sequence with a common ratio of ⅔, but for n=1, this expression gives us 27*⅔ which is not equal to 27.
Option D: [tex]27(2/3)^{(n-1)}[/tex] when evaluated for n=1 gives us 27*[tex](2/3)^{(1-1)}[/tex] = 27*1 = 27, which is the correct first term of our sequence. Applying this for subsequent terms also gives the correct sequence. Therefore, this is the correct expression for the nth term of the pattern.
Thus, the expression to find the nth term of the pattern is Option D: [tex]27(2/3)^{(n-1)}[/tex].
Write an equation for the perpendicular line in slope intercept form for (-3,5);y=1/3x-2
Hey there! :)
We're given the equation : y = 1/3x - 2 & coordinates : (-3, 5)
If we're looking for an equation for a perpendicular line, then we know that the slope of our new equation MUST be the negative reciprocal of the original slope. This means that you flip the slope and add or remove a negative. Example : slope = -2 ⇒ negative reciprocal slope = 1/2
Before using this information, we must first find the slope of our original equation.
Using slope-intercept form, which is : y=mx+b ; where m=slope, b=y-intercept
Ask yourself : which value is in the "m" spot and which value is in the "b" spot?
Using this, we now know that 1/3 is our slope, and -2 is our y-intervept.
So, our negative reciprocal is -3.
Use point-slope form to find the y-intercept, and coordinates (-3, 5)
Point-slope = y - y1 = m(x - x1)
y - 5 = -3(x - (-3))
Simplify.
y - 5 = -3(x + 3)
Simplify.
y - 5 = -3x - 9
Add 5 to both sides.
y = -3x + 4 ⇒ slope-intercept form
~Hope I helped! :)
Which of the following is a list of equivalent numbers?
A. 1.25,114,12.5%
B. 0.125,14,12.5%
C. 12.5,1212,125%
D. 1.25,114,125%
ANSWER
D. 1.25,1¼,125%
EXPLANATION
The number 1.25 can be rewritten as a fraction to obtain:
[tex]1 \frac{1}{4} [/tex]
We can covert 1.25 into percentage by multiplying by 100%
This implies that,
[tex]1.25 = 1.25 \times 100\% = 125\%[/tex]
Therefore the equivalent numbers are:
1.25,1¼,125%
The correct answer is D
I need help with solving this problem. A picture is attached! Ty
[tex]\bf \stackrel{\stackrel{\textit{net income}}{\textit{revenue}}}{f(x)}=9x^2-54x-144\implies \stackrel{\textit{revenue}}{0}=9x^2-54x-144 \\\\\\ 0=9(x^2-6x-16)\implies 0=x^2-6x-16 \\\\\\ 0=(x+2)(x-8)\implies x= \begin{cases} -2\\ 8&\checkmark \end{cases}[/tex]
since the units sold can't be negative, we can't use -2.
find the mean of these values 100, 200, 300, 400
250
100 + 200 + 300 + 400 = 1000
1000/4 = 250
Answer:
250
Step-by-step explanation:
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.
Lloyd's Bakery sold one customer 9 dozen chocolate cookies and 8 dozen oatmeal cookies for $110. The bakery also sold another customer 9 dozen chocolate cookies and 5 dozen oatmeal cookies for $89. How much do the cookies cost?
A dozen chocolate cookies cost $___ and a dozen oatmeal cookies cost $___
To describe the situation, we can write a system of equations. The cost of a dozen chocolate cookies and a dozen oatmeal cookies can be found by solving the system of equations.
Explanation:To write a system of equations to describe the situation, let's assign variables to the unknown quantities. Let's say the cost of a dozen chocolate cookies is x dollars and the cost of a dozen oatmeal cookies is y dollars.
The first equation represents the sale of 9 dozen chocolate cookies and 8 dozen oatmeal cookies for $110. This can be written as:
9x + 8y = 110
The second equation represents the sale of 9 dozen chocolate cookies and 5 dozen oatmeal cookies for $89. This can be written as:
9x + 5y = 89
We now have a system of two equations with two variables. We can solve this system using any method, such as substitution or elimination, to find the values of x and y.
Once we find the values of x and y, we can determine the cost of a dozen chocolate cookies and a dozen oatmeal cookies, respectively.
The number of frogs in a certain lake is inversely related to the number of snakes in the lake. If x represents the number of snakes in the lake, then y = 100/x represents the number of frogs in the lake. Describe the reasonable domain and range values. Graph the function.
1. Domain: The set of natural numbers / Range: The set of natural numbers
2. Shown below
Step-by-step explanation:1. Domain and Range.In this problem, we have that the number of frogs in a certain lake is inversely related to the number of snakes in the lake. Hence we are facing an Inverse Variation, so this means that if the number of snakes in the lake increases then the number of frogs in the lake decreases, because snakes eat frogs! The function that describes this is a rational function defined as:
[tex]y=\frac{100}{x}[/tex]
Where:
[tex]x: \ represents \ the \ number \ of \ snakes \ in \ the \ lake \\ \\ y: \ represents \ the \ number \ of \ frogs \ in \ the \ lake[/tex]
As you can see, [tex]x[/tex] is in the denominator, therefore [tex]x\neq 0[/tex]. Since we need to provide a reasonable domain, we say that the domain is the set of natural numbers and this doesn't include the number 0. Why aren't negative values included in the domain as well? Well, although negative values are included in the function, they aren't reasonable because [tex]x[/tex] represents the number of snakes and you always get positive numbers when counting things.
To get the range, let's take the inverse of this function, so:
[tex]y=f(x)=\frac{100}{x} \\ \\ \\ Interchange \ x \ and \ y: \\ \\ x=\frac{100}{y} \\ \\ \\ Isolate \ y: \\ \\ y=\frac{100}{x} \\ \\ f^{-1}(x)=\frac{100}{x}[/tex]
So the domain of [tex]f^{-1}(x)[/tex] is the range of our given function [tex]y=f(x)[/tex]. As you can see the inverse function is the same as our given function, then the range is the set of natural numbers as well.
2. Graph.First of all, we can define the pattern of the rational function as:
[tex]y=g(x)=\frac{1}{x}[/tex]
So our function will be:
[tex]f(x)=100g(x)=100\frac{1}{x}=\frac{100}{x}[/tex]
So the graph of [tex]f(x)[/tex] will be the same graph of [tex]g(x)[/tex] but it's been stretched vertically by a constant of 100, that is, each y-value is multiplied by 100. Also, since [tex]x \neq 0[/tex] and [tex]y \neq 0[/tex] then at [tex]x=0[/tex] there is a vertical asymptote and at [tex]y=0[/tex] there is an horizontal asymptote. Finally, the graph is shown below for [tex]x>0 \ and \ y>0[/tex], but remember: Whenever [tex]x \ and \ y[/tex] are natural numbers.
Answer:
Both the number of snakes and the number of frogs will be positive, so positive values are reasonable for the domain and range.
Step-by-step explanation: