Using a proof by contraposition we can see that the sum must be even.
How to use the proof by contraposition?Here we know that x and y are odd integers, then we can write thes two numbers as:
x = 2n + 1
y = 2m + 1
Where n and m are integers.
If we use the proof by contraposition, then we will assume that the outcome is also an odd number 2k + 1
Then we can write:
2n + 1 + 2m + 1 = 2k + 1
We can simplify this to get:
2n + 2m + 1= 2k
2(n + m) + 1 = 2k
So there we have an odd number is equal to an even number, that does not make sense, thus, the outcome of the sum can't be odd.
Learn moer about proof by contraposition
https://brainly.com/question/30426179
#SPJ1
given h(x)=2x^2-7x+4, find h(-7)
The output value of h(-7) in the function h(x) = 2x² - 7x + 4 is 151.
What is the output value of h(-7) in the given function?A function is simply a relationship that maps one input to one output.
Given the function in the question
h(x) = 2x² - 7x + 4,
Find h(-7)
To find h(-7), we substitute -7 for x in the equation of h(x) and simplify:
h(x) = 2x² - 7x + 4
Plug in x = -7
h(-7) = 2(-7)² - 7(-7) + 4
h(-7) = 2(49) + 49 + 4
h(-7) = 98 + 49 + 4
h(-7) = 151
Therefore,the output value of h(-7) is equal to 151.
Learn more about functions here: https://brainly.com/question/28783997
#SPJ1
a tree in lamar’s backyard has been growing for 4 1/2 years. the tree is 4/5 meters tall. what is the unit rate in meters per year? write your answer as a fraction or pixies number in simplest form.
As the tree is 4/5 meters tall the unit rate in meters per year is 8/45 or approximately 0.177 meters per year.
To find the unit rate in meters per year, we need to divide the height of the tree by the time it took to grow.
Height of the tree = 4/5 meters
Time taken to grow = 4 1/2 years = 9/2 years
Unit rate = Height/Time = (4/5)/(9/2) meters per year
To divide by a fraction, we can multiply by its reciprocal:
Unit rate = (4/5) * (2/9) meters per year
Unit rate = 8/45 meters per year
Therefore, the unit rate of the tree in meters per year is 8/45 or approximately 0.177 meters per year.
For more details regarding unit rate, visit:
https://brainly.com/question/29781084
#SPJ1
A caterer made a sandwich 6' 6" long for a party. by request 3' 8" were added to the sandwich. what was the total length of the sandwich?
The total length of the sandwich, given the addition to the sandwich, would be 10. 17 feet.
How to find the length of the sandwich ?In order to accurately add lengths measured in feet and inches, it is necessary to transform the unit type prior to adding them together. As an example, take a sandwich that originally measured 6 feet 6 inches long with 3 feet 8 inches being added into it.
To combine these measurements, we must first convert inches to feet or vice versa for optimal results. Let's begin with transforming the inches to feet:
6 feet 6 inches = 6 + ( 6 / 12 ) feet = 6. 5 feet
3 feet 8 inches = 3 + ( 8 / 12 ) feet = 3. 67 feet
Then add the lengths to become :
6. 5 feet + 3. 67 feet = 10. 17 feet
Find out more on length at https://brainly.com/question/18698394
#SPJ1
I NEED HELP ASAP PLEASE HELP!!!!!!
Aisha spent $120.94 to buy 2 wheelbarrows. The wheelbarrows both had the same price. How much did each wheelbarrow cost?
Eight cards with one letter on each card spell out the word SURPRISE. If you choose one card at random, what is the probability of this event?
P(R) =
what is true about the modeling equation that best fits the data?
• It captures the overall trends and patterns in the data points. It does not necessarily pass through every single data point but approximates the shape and trajectory of the data.
• It has parameters that can be adjusted to optimize how well the model fits the data. Things like coefficients, exponents, intercepts, etc. The good parameter values will depend on the specific data set and model.
• It has an appropriate level of complexity for the data. It does not overfit or underfit the data. An overfit model captures every detail and noise, while an underfit model fails to capture the key dynamics.
• Key features like slope, curvature, periodicity, thresholds, etc. in the data are properly represented in the model.
• It provides insights or useful information, not just a mathematical abstraction of the data. A good model opens the black box and provides interpretable knowledge about the system or phenomenon that generated the data.
• Predictions or extrapolations from the model align well with any additional data points beyond the original data set. The model generalizes purposefully.
• There are objective metrics that can evaluate how good a particular model is for a data set, like R2, mean absolute error, Akaike information criterion, etc. The best model will optimize such metrics.
5. Shanequa graphed a triangle with the coordinates R(-2, 1), S(2, 1) and T(0, -3). She wanted to graph
a dilation of the figure with a scale factor of 2 centered at the origin so she graphed the points
R'(-2, 2), S'(2, 2) and T'(0, -6).
a. Is AR'S'T' a dilation of ARST? Why or why not?
b. If Shanequa was wrong, correct her answer by giving the correct coordinates for AR'S'T'.
c. Shanequa's teacher said Shanequa had made a common mistake. Explain her mistake.
a) No , it is not a dilation
b) The coordinates are R'(-2, 2), S'(2, 2), and T'(0, -6)
c) Shanequa's mistake is that she incorrectly applied the dilation transformation
Given data ,
a)
In ARST, the coordinates of the original triangle are given by R(-2, 1), S(2, 1), and T(0, -3), and in AR'S'T', the coordinates are R'(-2, 2), S'(2, 2), and T'(0, -6)
So , the dilated triangle is R'(-2, 2), S'(2, 2), and T'(0, -6)
b)
The coordinates for AR'S'T' can be calculated by multiplying the original coordinates of ARST by the scale factor of 2, centered at the origin.
R'(-4, 2), S'(4, 2), and T'(0, -6)
c)
The dilation transformation was applied wrongly by Shanequa. The coordinates of the original points in a dilation centred at the origin with a scale factor of two should be multiplied by the scale factor to obtain the coordinates of the dilated points.
Hence , the dilation is solved
To learn more about dilation click :
https://brainly.com/question/13176891
#SPJ1
Solve the following system of linear equations 3x₁ + x₂ + 2x3= 6
4x₁ - 3x₂ + 5x3 = 8,
2x₁ + x₂ + 3x3 = 10
Okay, here are the steps to solve this system of linear equations:
3x1 + x2 + 2x3= 6
4x1 - 3x2 + 5x3 = 8,
2x1 + x2 + 3x3 = 10
1) Add 4 times the first equation to the second equation:
7x1 + 2x2 + 7x3 = 22
2) Add -2 times the first equation to the third equation:
5x1 - x2 + 5x3 = 12
3) Divide both sides by 5 to get:
x1 = 2
x2 = -2
x3 = 1
Therefore, the solution to the system is:
x1 = 2
x2 = -2
x3 = 1
Let me know if you have any other questions!
Consider the quadratic function f(x) = x2 – 5x + 12. Which statements are true about the function and its graph? Select three options. The value of f(–10) = 82 The graph of the function is a parabola. The graph of the function opens down. The graph contains the point (20, –8). The graph contains the point (0, 0).
The three (3) true statements about the quadratic function and its graph include the following:
A. The value of f(–10) = 82
B. The graph of the function is a parabola.
D. The graph contains the point (20, –8).
How to determine the true statements about this quadratic function?Generally speaking, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. For this quadratic function, the graph is a upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero.
Next, we would determine the other true statements about the graph of this quadratic function:
At point (-10, 82), we have:
Quadratic function, f(x) = 1/5 x² – 5x + 12
Quadratic function, f(x) = x²/5 – 5x + 12
Quadratic function, f(-10) = -10²/5 – 5(-10) + 12
Quadratic function, f(-10) = 82
At point (20, -8), we have the following:
Quadratic function, f(x) = 1/5 x² – 5x + 12
Quadratic function, f(x) = x²/5 – 5x + 12
Quadratic function, f(20) = 20²/5 – 5(20) + 12
Quadratic function, f(20) = -8
In conclusion, we can logically deduce that the graph of this quadratic function does not contain the point (0, 0) as shown in the image attached below.
Read more on quadratic function here: brainly.com/question/13159926
#SPJ1
Math: please answer this question very important for me, I’ll give brainliest if it’s correct!!
Q2. The average depth of the water in a port on a tidal river is 4 m. At low tide, the depth of the water is 2 m. One cycle is completed approximately every 12h.
(a) [4 marks] Find an equation of the depth, d(t) meters, with respect to the average depth, as a function of the time, t hours, after low tide, which occurred at 15: 00.
(b) [2 marks] Draw a graph of the function for 48 h after low tide.
5x+3y=4 -2x-8y=6 Which strategy can you use to eliminate a variable.
The strategy that could be used to eliminate a variable is:
Multiply the first equation by 2 and multiply the second equation by 5. Then, add the resulting equations
Solving system of linear equations using the elimination method: Determining a strategy to eliminateFrom the question, we are to determine a strategy that could be used to eliminate a variable in the given system of equations
From the given information,
The system of linear equations is
5x + 3y = 4
-2x - 8y = 6
To eliminate the variable x, we can multiply the first equation by 2 and then multiply the second equation by 5. After that, we will add the resulting equations
2 × [ 5x + 3y = 4
5 × [-2x - 8y = 6
10x + 6y = 8
-10x - 40y = 30
Add the resulting equations
10x + 6y = 8
+ (-10x - 40y = 30
------------------------
-34y = 38
This way we have eliminated x
Hence,
The above strategy can be used to eliminate a variable
Learn more on Elimination method here: https://brainly.com/question/28405823
#SPJ1
Given that sin theta = 15/17 with theta quadrant 1 , find cos 2theta
[tex]\textit{Double Angle Identities} \\\\ \cos(2\theta)= \begin{cases} \cos^2(\theta)-\sin^2(\theta)\\ 1-2\sin^2(\theta)&\leftarrow \textit{we'll use this one}\\ 2\cos^2(\theta)-1 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ 1-2\sin^2(\theta)\implies 1-2\left( \cfrac{15}{17} \right)^2\implies 1-2\left( \cfrac{225}{289} \right) \\\\\\ 1-\cfrac{450}{289}\implies \cfrac{289-450}{289}\implies -\cfrac{161}{289}[/tex]
I need help plssss i domt get it
Answer:
V = 5747.02 ft^3
Step-by-step explanation:
We're given the general formula for volume (V) and we must use this general formula and the information we're given to find the volume of the hemisphere.r stands for radius, which is 1/2 the distance between the center of a circle and the edge. The 14 in the diagram represents the radiusIn the formula r^3 means that the radius is cubed and thus 14 must be multiplied by itself three times (14 * 14 * 14)
Therefore, to solve for volume, we plug in 14 for r in the general formula:
[tex]V=2/3\pi (14)^3\\V=2/3\pi (14)(14)(14)\\V=2/3\pi *2744\\V=5747.020161\\V=5747.02[/tex]
Lastly, volume is always in units cubed (e.g., ft^3 or mi^3 since volume deals with the three-dimensional space of an object
For the rotation -757°, find the coterminal angle from 0° ≤ 0 < 360°, the
quadrant, and the reference angle.
To find the coterminal angle for a rotation of -757°, we can add or subtract multiples of 360° until we find an angle between 0° and 360° that is coterminal with -757°.
Adding 360° multiple times:
-757° + 360° = -397° (not coterminal)
-397° + 360° = -37° (not coterminal)
-37° + 360° = 323°
Therefore, the coterminal angle for -757° is 323°.
To find the quadrant in which 323° lies, we can note that 323° is between 270° and 360°, which means it lies in the fourth quadrant.
To find the reference angle for 323°, we can subtract 360° from 323° until we get an angle between 0° and 90°:
323° - 360° = -37° (not in the correct range)
-37° + 360° = 323°
So the reference angle for 323° is 37°.
The body temperatures in degrees Fahrenheit of a sample of adults in one small town are:
98 97.9 99 97.3 98.9 99.6 97.8 99.8 99.1 98.4 98.7 97.6 96.5
Assume body temperatures of adults are normally distributed. Based on this data, find the 95% confidence
interval of the mean body temperature of adults in the town. Enter your answer as an open-interval (i.e.,
parentheses) accurate to 3 decimal places. Assume the data is from a normally distributed population.
95% C.I. =
The 95% confidence interval of the mean body temperature of adults in the town is (97.584, 98.876)°F (open-interval notation), accurate to 3 decimal places.
What is the 95% confidence interval of the mean body temperature of adults in the town?To find the 95% confidence interval (C.I.) of the mean body temperature, we need to use the formula:
C.I. = x ± z*(σ/√n)
Where:
x is the sample mean
σ is the population standard deviation (unknown, but we can estimate it using the sample standard deviation)
n is the sample size
z is the critical value for the desired confidence level (95% in this case)
First, we need to calculate the sample mean, sample standard deviation, and sample size:
x = (98 + 97.9 + 99 + 97.3 + 98.9 + 99.6 + 97.8 + 99.8 + 99.1 + 98.4 + 98.7 + 97.6 + 96.5) / 13 = 98.23
s = √((1/(n-1)) * Σ(xi - x)²)
s = 1.339
n = 13
Next, we need to find the critical value, z, for a 95% confidence level. We can look this up in a standard normal distribution table or use a calculator. For a 95% confidence level, z = 1.96.
Now we can plug in the values into the formula to get the 95% confidence interval:
C.I. = 98.23 ± 1.96*(1.339/√13) = (97.584, 98.876)
Learn more about confidence intervals at: https://brainly.com/question/17097944
#SPJ1
$44,000 a 6% for 2 years for simple interest
The total interest that will accrue on the principal amount of $44,000 over a period of 2 years at a simple interest rate of 6% is $5,280.
The given problem involves calculating the amount of interest that will accrue on a principal amount of $44,000, which is invested at a simple interest rate of 6% for a period of 2 years.
To solve the problem, we can use the formula for simple interest, which is:
Interest = Principal x Rate x Time
In this case, the principal amount is $44,000, the rate of interest is 6%, and the time period is 2 years. So, plugging in these values, we get:
Interest = $44,000 x 0.06 x 2
Interest = $5,280
Unlike compound interest, simple interest is calculated only on the principal amount, and not on the accumulated interest. This means that the interest earned each year remains constant, which can make it easier to calculate and understand compared to compound interest.
To learn more about interest click on,
https://brainly.com/question/28332019
#SPJ1
Please help
Question in image
Answer:
C 86
Step-by-step explanation:
You want the measure of the angle marked x° where chords cross. The angle marked x° intercepts an arc of 104° and one that is unmarked. The remaining arcs of the circle are marked 111° and 77°.
Measure of xThe measure of angle x is half the sum of the arcs it intercepts. One of those is given as 104°. The other will be ...
360° -111° -104° -77° = 68°
Then the value of x is ...
(104 +68)/2 = 172/2 = 86
The value of x is 86, choice C.
<95141404393>
#9 75% of 1
#10 50% of 350 is
#11 What percent of 450 is 50
#9 75% of 1.
#10 50% of 350.
#11
To find out what percent of 450 is 50, we can set up a proportion
x/100 = 50/450
Simplifying the right-hand side of the equation by dividing both numerator and denominator by 50 gives
x/100 = 1/9
Multiplying both sides by 100 gives
x = 100/9
Hence, 50 is approximately 11.1% of 450.
To know more about percent here
https://brainly.com/question/246842
#SPJ1
Patel is solving 8x2 + 16x + 3 = 0. Which steps could he use to solve the quadratic equation? Select three options. 8(x2 + 2x + 1) = –3 + 8 x = –1 Plus or minus StartRoot StartFraction 5 Over 8 EndFraction EndRoot x = –1 Plus or minus StartRoot StartFraction 4 Over 8 EndFraction EndRoot 8(x2 + 2x + 1) = 3 + 1 8(x2 + 2x) = –3
Answer:
Step-by-step explanation:
Patel is solving 8x2 + 16x + 3 = 0. Which steps could he use to solve the quadratic equation? Select three options. 8(x2 + 2x + 1) = –3 + 8 x = –1 Plus or minus StartR…
✓ Answer:8(x2 + 2x) = –3 8(x2 + 2x + 1) = –3 + 8 x = –1 Plus or minus StartRoot StartFraction 5 Over 8 EndFraction EndRootStep-by-step explanation:S…
Is anyone here looking for a personal tutor?????
meAnswer: me
Step-by-step explanation: I need help
there are 105 days until basketball season starts. Melanie says the season starts in 15 weeks. Is this reasonable? Use estimation to justify your answer.
Answer:
Yes, Melanie's statement that the basketball season starts in 15 weeks is reasonable.
To see why, we can estimate the number of days in 15 weeks. Since 1 week is equal to 7 days, 15 weeks would be approximately equal to:
15 weeks x 7 days/week = 105 days
Plot the numbers -1 1/2 and 2 3/4 on the number line below.
For the second number 2 3/4 you will find the 2 points, to the right from 0. Now count three small points to the left to get the 3/4 part
How to solveTo locate the value of -1 1/2 on the number line, first locate the point denoting -1 to the left of 0. Progressing to the second minor point on the right immediately after -1 will indicate this position.
For the second figure of 2 3/4, identify the point at 2 by moving to the right from 0. An additional count of three minor points towards the left reveals where the fraction of 3/4 is located. The exact spot is depicted here.
Read more about number lines here:
https://brainly.com/question/24644930
#SPJ1
A sports statistician determined that the probability of a certain rugby team winning its next match is 11/23. Find the odds against the team winning its next match.
The probability of the odds of the rugby team winning its next match is
12 to 11.
We have,
The probability of an event happening is defined as the ratio of the number of favorable outcomes to the total number of outcomes.
The odds against an event happening are defined as the ratio of the number of unfavorable outcomes to the number of favorable outcomes.
If the probability of the rugby team winning its next match is 11/23, then the probability of the team losing its next match is:
= 1 - 11/23
= 12/23.
The odds against the rugby team winning its next match are the ratio of the number of unfavorable outcomes (losing the match) to the number of favorable outcomes (winning the match), which is:
12/23 : 11/23
We can simplify this ratio by dividing both the numerator and denominator by 11:
(12/23) ÷ (11/23) : (11/23) ÷ (11/23)
Simplifying.
12/11 : 1/1
Therefore,
The probability of the odds of the rugby team winning its next match is
12 to 11.
Learn more about probability here:
https://brainly.com/question/14099682
#SPJ1
a bookshelf holds 24 mysteries, 12 nonfiction, and 4 biographies. what is the probability of selecting a mystery book, then without replacing it, selecting a non-fiction book?
The probability of selecting a mystery book, then without replacing it, selecting a nonfiction book is approximately 0.1462.
How to determine the probability of selecting a mystery book, then without replacing it, selecting a non-fiction bookOn the bookcase, there are 24 + 12 + 4 = 40 volumes.
The odds of picking a mystery book on the first draw are 24/40.
There are 39 books remaining after the first is drawn, including 12 nonfiction works. Because the initial book was never replaced, there are currently only 11 nonfiction volumes left.
Given that a mystery book was chosen without replacement in the first draw, the probability of selecting a nonfiction book in the second draw is 12/39.
When we multiply these probability together, we get:
P(nonfiction after mystery, not replacing) * P(mystery, not replacing) = (24/40) * (12/39) = 0.14615...
P(nonfiction after mystery, not replacing) * P(mystery, not replacing) = 0.1462
So, the probability of selecting a mystery book, followed by a nonfiction book without replacing it is roughly 0.1462.
Learn more about probability at https://brainly.com/question/25870256
#SPJ1
Winnie and Vanessa Co Ltd. borrowed an amount of money (P) from a rural bank in Miotso at i% p.a simply interest. The company paid A amount of money as well as the total amount of interest (I) to the bank at the end of n- years. Show that, p= A(1+in)-1
The amount of money borrowed by the company from the rural bank is:
P = [tex]A(1 + in)^{-1}[/tex]
What is Amount?
Amount is a term used to describe the total quantity or value of something. It can refer to a quantity of a physical substance or object, such as the amount of water in a bottle or the amount of money in a bank account.
Let P be the amount of money borrowed by Winnie and Vanessa Co Ltd from the rural bank at an interest rate of i% per annum.
The simple interest on this amount after n years will be:
I = P * i * n
The total amount to be paid by the company after n years will be:
A = P + I = P + P * i * n
We can simplify this expression by factoring out P:
A = P(1 + i * n)
Dividing both sides of this equation by (1 + i * n), we get:
P = A / (1 + i * n)
Therefore, the amount of money borrowed by the company from the rural bank is:
P = [tex]A(1 + in)^{-1}[/tex]
This is the desired result.
Learn more about Amount
https://brainly.com/question/25109150
#SPJ1
Suppose that prices of a certain model of new homes are normally distributed with a mean of $150,000. Use the 68-95-99.7 rule to find the percentage of buyers who paid:
between $150,000 and $156,000 if the standard deviation is $2000.
The percentage of buyers who paid between $150,000 and $156,000 if the standard deviation is $2000 is 49.9%
Calculating the percent of data of values from the the z-scoresFrom the question, we have the following parameters that can be used in our computation:
Mean = 150000
Standard deviation = 2000
Scores = from $150,000 and $156,000
So, we have
z = (150000 - 150000)/2000 = 0
z = (156000 - 150000)/2000 = 3
This means that it is between a z-score of 0 and a z-score of 3
This is represented as
Probability = (0 < z < 3)
Using a graphing calculator, we have
Probability = 0.49865
Express as percentage
Probability = 49.9%
Hence, the probability is 49.9%
Read mroe about z-scores at
brainly.com/question/25638875
#SPJ1
For a certain set of five numbers, the mean of all but the largest number is 80 and the
mean of all but the smallest number is 90. What is the range of the set of five numbers?
Thus, the range of the set of five numbers (a,b,c,d,e) is found as : 40.
Explain about the range of set:The difference in between highest and lowest numbers in a piece of data is known as the range. To locate it, sort the data first from least to greatest. Then take the difference between the least and largest number in the set.
The range is a straightforward assessment of the variation in values within a dataset. Simply take the difference between the two values, ignoring the others, to determine the range.
Let the 5 numbers be: (a,b,c,d,e)
a < b < c < d < e
Mean = sum of all observation / total number of observations
Equation for the mean except the largest number.
(a+b+c+d)/4 = 80
Now, Equation for the mean except the smallest number.
(b+c+d+e)/4 = 90
On simplifying each:
a + b + c + d = 320 ...eq 1
b + c + d + e = 360 ..eq 2
Range = max - min
Subtract eq 1 from eq 2
e - a = 40
Thus, the range of the set of five numbers (a,b,c,d,e) is found as : 40.
Know more about the range of set:
https://brainly.com/question/24326172
#SPJ1
Please help with question #2
Answer:
50 in
Step-by-step explanation:
Split into 4 shapes. One on each side with 5 in x 2 in, one on the top with 12 in x 2 in, and one in the middle with 3 in x 2 in. We add all together:
2(5×2)+(12×2)+(3×2) = 50 in
Let = {1, 2, 3, 4, 5} be a set. Consider the functions = {(1,3), (2,5), (3,3), (4,1), (5,2)} and = {(1,4), (2,1), (3,1), (4,2), (5,3)} From into . (i) Determine the range of and (ii) Find the composition functions ∘ and ∘ .
g ∘ f is the function {(1,2), (2,1), (4,2), (5,3)}. And the range of f is {1, 2, 3, 5}, and the range of g is {1, 2, 3, 4}.
How to solve the problem?
(i) To determine the range of a function, we need to find all the possible output values of the function. In other words, we need to find the set of all second elements in the ordered pairs of the function. For the function f, the set of all second elements is {3, 5, 1, 2}. For the function g, the set of all second elements is {4, 1, 2, 3}. Therefore, the range of f is {1, 2, 3, 5}, and the range of g is {1, 2, 3, 4}.
(ii) To find the composition functions f ∘ g and g ∘ f, we need to apply the functions in the correct order. The composition function f ∘ g means that we first apply g and then apply f to the result. The composition function g ∘ f means that we first apply f and then apply g to the result.
For f ∘ g:
First, we apply g to the domain of f.
(1,3) is mapped to (4,2) by g
(2,5) is mapped to (1,4) by g
(3,3) is mapped to (1,4) by g
(4,1) is mapped to (2,1) by g
(5,2) is mapped to (3,1) by g
Now we apply f to the result of g.
(4,2) is mapped to 2 by f
(1,4) is mapped to 4 by f
Therefore, f ∘ g is the function {(1,4), (4,2)}.
For g ∘ f:
First, we apply f to the domain of g.
(1,4) is mapped to 3 by f
(2,1) is mapped to 5 by f
(3,1) is mapped to 3 by f
(4,2) is mapped to 1 by f
(5,3) is mapped to 2 by f
Now we apply g to the result of f.
3 is mapped to (1,2) by g
5 is mapped to (2,1) by g
1 is mapped to (4,2) by g
2 is mapped to (5,3) by g
Therefore, g ∘ f is the function {(1,2), (2,1), (4,2), (5,3)}.
Note that the composition functions f ∘ g and g ∘ f are not the same. In general, composition of functions is not commutative, i.e., f ∘ g is not equal to g ∘ f.
To know more about range visit :-
https://brainly.com/question/26098895
#SPJ1