As below it is described what is exponential function with some examples.
What is Function ?A function is simply a “mapping” between 2 sets of objects (usually numbers but they could really be anything) such that, every object in the first set (the domain) is mapped to a single object in the second set. What you usually see as examples of functions (e.g. f(x)=x²) actually define the relationship between the sets. This is what I mean by “mapping”. Often, we don’t explicitly state what the sets are, because it’s understood from the context, but technically, it’s not a function unless we state the sets. E.g., instead of saying f(x)=x², we should say f:x↦x²|x∈ℂ (pronounced “f maps x to x² such that x is a complex number”), meaning that we map the complex numbers to themselves using the rule that we square each input. You can define it differently, so that, for example, we map complex to reals. However, and this is important, that would be a different function, even though the “rule” for turning inputs into outputs is the same.
Exponential Function:
Exponential functions are functions of the form f(x) = b^x where b is a constant.
The real mathematical importance of exponential functions is in their being proportional to their derivatives meaning the bigger x is, the steeper the slope of the function. This means they grow extremely fast: exponentially fast.
A common example of exponential growth is a bacterial population. Since bacteria reproduce rapidly by mitosis, for every bacterium in a culture, there will, before long be two bacteria. Thus, if I have 10 bacteria to start, one minute later I will have 20, and in another minute, each of those twenty will have reproduced and I will have 40; one minute later the same will have happened again and before long, the culture will be overrun with bacteria. The more bacteria there are, the more bacteria there are to make even more bacteria the next minute.
All exponential functions are proportional to their derivatives, but the function f(x) = e^x (where e = 2.718281828459…) is actually equal to its derivative meaning the y value of the function at a particular x value is equal to the slope of the curve at that same x value. Thus when dealing with exponential functions in calculus, e becomes a very natural number to use as a base. This is why e is such a big deal and why teachers try to introduce the number early on in Algebra 2, even though it doesn’t have much use at that point in one’s mathematical education.
When plotting an exponential curve by hand, one ought to simply pick whole number x values and multiply the base by itself an x number of times to calculate the corresponding y value. This will give a rough plot of the graph of the curve, although a calculator is requires to plot the points not on whole number x values. The curve of simple exponential functions is always increasing and concave up, or vice versa for b^-x. For a positive exponential function like f(x) = e^x, as x approaches ∞, f(x) approaches ∞, and as x approaches -∞, f(x) approaches 0. The curve has no absolute maximum or minimum value and has a horizontal asymptote at y = 0.
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When a number is divided by 9, and the remainder is 6. What is the
remainder when six times this number is divided by 9?
Answer:
9x + 6
Step-by-step explanation:
SOLUTION GIVEN: a number is divided by 9 and the remainder is 6,
TO DETERMINE: The remainder when six times the number is divided by 9
EVALUATION: Here it is given that the number is divided by 9 & the remainder is 6
We know that Dividend = Divisor × Quotient + Remainder
By the given Divisor = 9
Remainder = 6
Let quotient = x
Then the number = Dividend = 9x + 6
Which graph represents the function f(x)=4x?
Responses
Answer:
slope = 4
points on graph = (0,0), (1,4)
Step-by-step explanation:
line going through quadrants 1 and 3. Straight line.
Car makers (Toyota, Ford, Honda, Hyundai, etc.) often provide low-interest financing to attract people to buy: 3%, 2%, 1%, and even 0%. A 1% loan for 4 years ends up costing about 2% of the original loan. How much total interest is paid for a $20,000 loan?
Using the formula of simple interest, the total interest paid on a $20000 loan at rate of 2% over 4 years is $1600
Simple InterestSimple interest is calculated with the following formula: S.I. = P × R × T, where P = Principal, R = Rate of Interest in % per annum, and T = Time, usually calculated as the number of years. The rate of interest is in percentage r% and is to be written as r/100.
Principal: The principal is the amount that initially borrowed from the bank or invested. The principal is denoted by P.Rate: Rate is the rate of interest at which the principal amount is given to someone for a certain time, the rate of interest can be 5%, 10%, or 13%, etc. The rate of interest is denoted by R.Time: Time is the duration for which the principal amount is given to someone. Time is denoted by T.Amount: When a person takes a loan from a bank, he/she has to return the principal borrowed plus the interest amount, and this total returned is called Amount.The formula is given S.I = P(1 + RT)
Data;
P = 20000r = 2%t = 4 yearsS.I = 20000(1 + 0.02 * 4)
S.I = 21600
The total interest paid is 21600 - 20000 = 1600
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Which equation models the line on the graph?
Answer:
C) y - 2 = 2 ( x - 3 )
Step-by-step explanation:
This is point-slope form btw. Hope it helps!
Graph
50 points
Marking brainliest
Answer:
[tex]\textsf{Vertex form}: \quad f(x)=(x-4)^2+2[/tex]
[tex]\textsf{Standard form}: \quad f(x)=x^2-8x+18[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{5.6 cm}\underline{Vertex form of a quadratic equation}\\\\$y=a(x-h)^2+k$\\\\where:\\ \phantom{ww}$\bullet$ $(h,k)$ is the vertex. \\ \phantom{ww}$\bullet$ $a$ is some constant.\\\end{minipage}}[/tex]
From inspection of the given graph:
vertex = (4, 2)y-intercept = (0, 18)Substitute the vertex and point (0, 18) into the vertex formula and solve for a:
[tex]\begin{aligned}y&=a(x-h)^2+k\\\implies 18&=a(0-4)^2+2\\18&=a(-4)^2+2\\18&=16a+2\\16a&=16\\a&=1\end{aligned}[/tex]
Substitute the found value of a and the vertex into the formula:
[tex]\implies y=1(x-4)^2+2[/tex]
[tex]\implies f(x)=(x-4)^2+2[/tex]
To write the equation in standard form, expand the brackets:
[tex]\implies f(x)=(x-4)^2+2[/tex]
[tex]\implies f(x)=(x-4)(x-4)+2[/tex]
[tex]\implies f(x)=x^2-8x+16+2[/tex]
[tex]\implies f(x)=x^2-8x+18[/tex]
y = 5x + 4
y = 5x - 2
One solution
No solution
Infinite solution
Answer: No solution
Step-by-step explanation:
Since both equations are equal to y you can set them equal to each other
5x+4=5x-2
0=-6
zero does not equal -6 which is that there is no solution.
Answer:
Step-by-step explanation:
the answer is 2 1/2 miles
Evaluate the expression: −(8 − 12) + 60 + (−4)2.
Answer: 80
Step-by-step explanation:
⇒ −(8 − 12) + 60 + (−4)²
⇒ -(-4) + 60 + 16
⇒ 4 + 60 + 16
⇒ 80
Write an equation of the line that passes through (8,9) and is parallel to the line y = -x-2.
An equation of the parallel line is
Answer:
y - 9 = -1 (x-8)Step-by-step explanation:
y - y₁ = m (x - x₁) would be your formula, since it's parallel you already have your slope, -x (which also equals 1 so technically -1). You have a point (8,9) and a slope -1, all you need to do is plug it in the formula.
Hope this helped!!
Let a,b,c be positive real numbers. Prove the inequality
[tex]\dfrac{(a+b)^2}{c} +\dfrac{c^2}{a} \geq 4b.[/tex]
If a,b,c are positive real numbers, the inequality [tex]\frac{(a+b)^{2} }{c} +\frac{c^{2} }a[/tex] [tex]\geq 4b[/tex] is not correct because it gives the value a³+2a²b+ab²≥4abc
How to find the solution set of an inequality?
Inequality is a mathematical statement showing that two quantities are not equal.
The given inequality is [tex]\frac{(a+b)^{2} }{c}[/tex] +[tex]\frac{c^{2} }{a}[/tex][tex]\geq[/tex]4abc
Simplifying the inequality we have [tex]\frac{(a+b)(a+b)}{c}[/tex] +[tex]\frac{c^{2} }{a }[/tex] [tex]\geq[/tex]4b
This implies that: [tex]\frac{a^{2}+ab+ab+b^{2} }{c}[/tex] +[tex]\frac{c^{2} }{a}[/tex] [tex]\geq[/tex] 4b
Taking the LCM,
[tex]\frac{a^{2}+2ab+b^{2} }{c}[/tex] +[tex]\frac{c^{2} }{a}[/tex] [tex]\geq[/tex]4b
Simplifying further
[tex]\frac{a^{3}+2a^{2}b+ab^{2}+c^{3} }{ac}[/tex] [tex]\geq[/tex]4b
The expression gives that: a³+2a²b+ab²≥4abc
Comparing the values of a,b,c on both sides of the inequality, the inequality is wrong.
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The ratio of a to b is constant, and a = 9 when b = 6. What is the value when a = 2? Round your answer to the nearest hundredths, if necessary
If ratio of a to b is constant, and a = 9 when b = 6 then value of b is 4/3 when a is 2.
What is Ratio?A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.
Given,
The ratio of a to b is constant,
a = 9 when b = 6.
We need to find value of b when a=2.
Let us consider the value be x.
Let us form an equation
9/6=2/x
Apply cross multiplication
9x=12
Divide both sides by 9
x=12/9=4/3
Hence, if ratio of a to b is constant, and a = 9 when b = 6 then value of b is 4/3 when a is 2.
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I need help with this please help me
Answer:
67.38
Step-by-step explanation:
134.76/2 = 67.38
May I please have brainliest for this? Thank you!
I need to show work
#1. (-2/7) + (-4/7)
#2. 5.4+ (-8.6)
Answer:
#1. -6/7
#2. -3.2
Step-by-step explanation:
#1. (-2/7) + (-4/7)
You are adding two negative numbers.
When you add two positive numbers or two negative numbers, just ignore the signs and add the numbers as positive numbers. Then the answer has the same sign as the original numbers.
Step 1. Ignore the signs and make both numbers positive.
Add 2/7 and 4/7.
2/7 + 4/7 = 6/7
Step 2. Since the two numbers are originally negative, the answer is negative.
(-2/7) + (-4/7) = -6/7
#2. 5.4 + (-8.6)
When you add a positive number and a negative number, first make both numbers positive. Then, subtract the smaller positive number from the larger positive number. Finally, the answer has the same sign as the original number that corresponds to the larger positive number.
Here we do it step by step.
5.4 + (-8.6)
Step 1. Make both numbers positive. The numbers now are:
5.4 and 8.6
Step 2. Subtract the smaller positive number from the larger positive number.
8.6 - 5.4 = 3.2
Now look at the two positive numbers. 5.4 and 8.6. The larger number is 8.6. The original number that 8.6 came from is -8.6, a negative number, so the answer is negative.
5.4 + (-8.6) = -3.2
which property justifies the step?
2x +3 = -5x -1
7x + 3 = -1
A. distrubutive property
B. addition property of equality
c. commutative property
D. subtraction property of equality
The required property justifies the step is an addition property of equality.
What is mathematical property?
The rules governing how numbers relate to and interact with one another are known as mathematical properties. There are four fundamental properties. i.e. distributive, addition, commutative and subtraction.
Given, 2x +3 = -5x -1
Applying addition property of equality,
Adding 5x to both sides, we get
7x + 3 = -1
Hence, the required property justifies the step is an addition property of equality.
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WILL GIVE BRAINLIEST Suppose that the functions q and r are defined as follows.
q (x) = x^2 + 7
r (x) = [tex]\sqrt{x+4}[/tex]
Find the following:
(r *q) (5) =
(q *r) (5) =
The values of the composite functions are
(r * q)(5) = 96
(q * r)(5) = 96
How to evaluate the composite functions?From the question, we have the following functions that can be used in our computation:
q(x) = x² + 7
r(x) = √(x + 4)
The composite functions are given as
(r * q)(5) and (q * r)(5)
The base functions of the above composite functions are
(r * q)(x) and (q * r)(x)
And they are calculated using
(r * q)(x) = r(x) * q(x)
(q * r)(x) = q(x) * r(x)
Substitute the known values in the above equations
So, we have the following equations
(r * q)(x) = (x² + 7) * (√(x + 4))
(q * r)(x) = (√(x + 4)) * (x² + 7)
Substitute 5 for x
(r * q)(5) = (5² + 7) * (√(5 + 4))
(q * r)(5) = (√(5 + 4)) * (5² + 7)
Evaluate the equations
(r * q)(5) = 96
(q * r)(5) = 96
Hence, the solution of the composite functions is 96
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Graph the following equation using a table of values,
y=x+10
Answer:
The table in the attached figure N 1
The graph in the attached figure n 2
Step-by-step explanation:
we have
x+2=y
we know that
To graph a linear equation the best way is plot the intercepts
Find the y-intercept
The y-intercept is the value of y when the value of x is equal to zero
For x=0
0+2=y
y=2
The y-intercept is the point (0,2)
Find the x-intercept
The x-intercept is the value of x when the value of y is equal to zero
For y=0
x+2=0
x=-2
The x-intercept is the point (-2,0)
Find additional points to create a table (is not necessary for plot the line)
For x=1
1+2=y
y=3
For x=2
2+2=y
y=4
The table in the attached figure N 1
Graph the linear equation ------> plot the intercepts a joined the points
see the attached figure N 2
Find the value of c guaranteed by the Mean Value Theorem for Integrals for the function 2x+1 on the interval [3,6].
The value of c is 0 guaranteed by the Mean Value Theorem for Integrals for the function 2x+1 on the interval [3,6].
What is Mean Value Theorem?Mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.
Given.
Function is 2x+1 on the interval [3,6]
f(x)=2x+1
f(x) continuous on [3, 6] and differentiable on (3, 6) so the MVT applies here.
f(3)=7
f(6)=13
Average rate of change of f on [3,6]
= (13-7) / (6-3) = 6/3=2
To find the c-value guaranteed by the MVT, such that the instantaneous rate of change when x = c is 2, we differentiate, set f'(x) = 2, and solve:
f'(x) = 2 and x = 1.
c = 0 is the only c-value in the interval at which the instantaneous rate of change = the average rate of change.
Hence, the value of c is 0 guaranteed by the Mean Value Theorem for Integrals for the function 2x+1 on the interval [3,6].
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I need help please help me with this
Answer:
120
Step-by-step explanation:
MNT is equal to RQT.
That means R equal to M, T equal to T, equal to N.
Q also equals 180-(T+R)
Q=180-(35+25)
Q=180-60
Q=120
Select a newspaper or magazine article that involves a statistical study and write a paper answering these questions. 1.Is this study descriptive or inferential? Explain your answer. 2.What are the variables used in the study? In your opinion, what level of measurement was used to obtain the data from the variables? 3.Does this article define the population? If so, how is it defined? If not, how could it be defined? 4.Does the article state the sample size and how the sample was obtained? If so, determine the size of the sample and explain how it was selected. If not, suggest a way it could have been obtained. 5.Explain in your own words what procedure (survey, comparison of groups, etc.) might have been used to determine the studys conclusions. 6.Do you agree or disagree with the conclusions? State your reason.
There are 300 students at Josefa Espinoza School.
reported viral illnesses very ... so the
in charge of the health of the Wisconsin area health department,
The study was in total in a sample of 60
According to the question. A journal article that involves a statistical study.
find:
a) The study descriptive or inferential
Yes, know the study is infrential according to the obtained from
Data
find :-) The Variable used in study. low,
과 a continuous variable. is size and weight
Know The sample is varied and representative
Solve the system of equations.
2/3 x - y/2 = 17/6
x/4 - 5/8 y = 5/8
Answer:
x = 3y/4 + 17/4 and
x = 5y/2 + 5/2
Step-by-step explanation:
hope this help
Simplify the Expressions Drag and drop the matching
The simple form of all the expressions are:
-3 + 4m + (-7) - 6m = -2m -10
6x + (-1) - 3x + 12 = 3x + 11
3xy +2z - 16 + 6xy = 9xy +2z - 16
-4x - 8 + 2x + (-3) - 5x = -7x - 11
3m² - m + 7 + 3m² + 1 = 6m² - m + 8.
What is the simple form of the polynomial?
f(x) = anxn + anxn+1 +... + a2x2 + a1x + a0 is the formula for a polynomial. The greatest power of x in an expression is the polynomial's degree. Polynomials of degree 0, 1, 2, 3, and 4 are constant (non-zero) polynomials, linear polynomials, quadratics, cubics, and quartics, respectively.
We have,
-3 + 4m + (-7) - 6m
6x + (-1) - 3x + 12
3xy +2z - 16 + 6xy
-4x - 8 + 2x + (-3) - 5x
3m² - m + 7 + 3m² + 1
The simple form of the expressions:
-3 + 4m + (-7) - 6m
= -2m -10
6x + (-1) - 3x + 12
= 3x + 11
3xy +2z - 16 + 6xy
= 9xy +2z - 16
-4x - 8 + 2x + (-3) - 5x
= -7x - 11
3m² - m + 7 + 3m² + 1
= 6m² - m + 8
Hence, the simple form of all the expressions are:
-3 + 4m + (-7) - 6m = -2m -10
6x + (-1) - 3x + 12 = 3x + 11
3xy +2z - 16 + 6xy = 9xy +2z - 16
-4x - 8 + 2x + (-3) - 5x = -7x - 11
3m² - m + 7 + 3m² + 1 = 6m² - m + 8
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You have $20 to spend and apples cost $1 per apple. The independent variable is apples. What is the Range and Domain?
The Range and Domain for the apples as per the given information is (1, 20) and (1, 20).
What is meant by range?The difference between the greatest and lowest values of a set of integers is defined as its range. To find it, subtract the distribution's lowest number from its highest.
Because the term "range" can have multiple meanings, it is recommended that it be defined the first time it is used in a textbook or article. When older books use the term "range," they usually indicate what is currently known as the codomain. More recent works, if they use the phrase "range," usually use it to refer to what is now known as the picture. A number of current books avoid using the word "range" entirely to avoid confusion.
Given,
We need to spend $20 for each apple
And also given that,
The cost of each apple is $1
Here the independent variables are apples
Range=(1, 20)
Domain=(1, 20)
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Find the midpoint of the line segment from (1,-3) to(-5,2)
Midpoint of the line segment from (1,-3) to(-5,2) is (-2,-1/2).
What is straight line?A straight line is just a line with no curves. So, a line that extends to both sides to infinity and has no curves is called a straight line.
The midpoint of a line segment is a point that lies exactly halfway between two points
The formula for midpoint (x,y)=(x₁+x₂/2,y₁+y₂/2)
We need to find midpoint of the line segment from (1,-3) to(-5,2)
Midpoint=(1-5/2,-3+2/2)
=(-4/2, -1/2)
=(-2,-1/2)
Hence, midpoint of the line segment from (1,-3) to(-5,2) is (-2,-1/2).
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Describe and correct the error in writing an equation of the line that passes through (1, 3) and is parallel to the line y=1/2x+2
The equation of the line that is parallel to the line y = 1/2x + 2 is: y = 1/2x + 5/2.
How to Write the Equation of Parallel Lines?Recall that if two lines are parallel, they will have the same slope (m).
Slope of a line (m) = change in y / change in x.
The slope (m) of the equation, y = 1/2x + 2, that represents a line is 1/2.
This means that the line that is parallel to it will also have a slope (m) that is equal to 1/2.
Find the y-intercept by substituting m = 1/2 and (x, y) = (1, 3) into y = mx + b:
3 = 1/2(1) + b
3 = 1/2 + b
3 - 1/2 = b
5/2 = b
b = 5/2
To write the equation of the parallel line, substitute m = 1/2 and b = 5/2 into y = mx + b:
y = 1/2x + 5/2.
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at the zoo 1/3 of the animals are mammals. 1/4 of the mammals are elephants. what fractuon of the animals at the zoo are elephants
Hence, the fractuon of the animals at the zoo are elephants is [tex]\frac{7}{12}[/tex].
What is the fraction?
A fraction defined as the part of the whole thing.
Fractions help to distribute and judge the numbers easily and make the calculation faster.
Here given that,
At the zoo [tex]\frac{1}{3}[/tex] of the animals are mammals [tex]\frac{1}4}[/tex] of the mammals are elephants.
So, the fraction of the animals at the zoo are elephants is
[tex]\frac{1}3}+\frac{1}{4}=\frac{4+3}{12}\\\\=\frac{7}{12}[/tex]
Hence, the fractuon of the animals at the zoo are elephants is [tex]\frac{7}{12}[/tex].
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Q4.
The 5th term of an arithmetic series is 4k, where k is a constant.
The sum of the first 8 terms of this series is 20k + 16
(a) (i) Find, in terms of k, an expression for the common difference of the series.
(ii) Show that the first term of the series is 16 - 8k
Given that the 9th term of the series is 24, find
(b) the value of k,
(c) the sum of the first 20 terms.
Answer:
check the attached file
Step-by-step explanation:
Solve the system of linear equations by graphing. y = 3/4x-4 y= -1/2x+11
Taking into account the definition of a system of linear equations, the solution of the linear equation is (12,5).
System of linear equationsA system of linear equations is a set of two or more equations of the first degree, in which two or more unknowns are related.
Solving a system of equations consists of finding the value of each unknown so that all the equations of the system are satisfied.
This caseIn this case, the system of equations to be solved is
y = 3/4x-4
y= -1/2x+11
There are several methods to solve a system of equations, it is decided to solve it using the graphical method. This method consists of representing the graphs associated with the equations of the system to deduce its solution. The solution of the system is the point of intersection between the graphs since the coordinates of said point satisfy both equations.
In this case, graphing both equations, it is obtained that the point of intersection, and therefore the solution, is (12,5).
The graph of the system of equations is attached.
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The monthly water bill is a function of the number of gallons used. The cost is $11.70 for 10,000 gal or less. Over 10,000 gal, the cost is $11.70 plus $1 for each 1000 gal over 10,000 with any fraction of 1000 gal charged at a fraction of $1. On what interval is the cost constant? On what interval is the cost increasing?
Answer:
Interval for constant cost = [0, 10,000] same as 0 <= x <= 10,000
Interval for increasing cost = (10,000, ∞) same as 10,000 < x < ∞
Be careful! note the use of different types of parentheses for each answer part as explained below
Step-by-step explanation:
We are only being asked for the two intervals, not the function definition itself which would have made it a little bit more work. A function has inputs and outputs. Here the input is gallons of water and output is $ cost.
Normally when asked for intervals we use a variable. Let's call it x which represents the gallons of water used
Since there is a base rate of $11.70 whether or not any water is consumed as long as it is less than or equal to 10,000 gallons we can state that over the interval 0 to 10000 (inclusive), the cost is constant
We can represent this interval in two ways
0 ≤ x ≤ 10,000
Another way of representing this interval is what we call interval notation. In interval notation the same thing can be expressed as:
[0, 10000]
Here 0 is referred to as the lower limit or bound(minimum) and the 10,000 is referred to as the upper limit or upper bound
Note the use of square brackets. This means that both upper and lower limits are included in the function. Such an interval is also called a closed interval.
After 10,000 gallons, the cost keeps increasing. There is no upper limit on the usage of water. So the lower limit is 10,000 (but does not include that number) and upper limit is infinity
We represent this interval as
10000 < x < ∞
Note that both upper and lower limits are excluded since ∞ is not realistic.
In interval notation this is represented as
(10000, ∞)
Here we are using regular brackets(parentheses) because this interval does not include lower and upper limits
This kind of interval is known as an open interval
Looks like the purpose of this question is to make sure what you understand in terms of interval notation and possibly open and closed intervals. The only relevant number there is the 10,000 gallons which represents the upper bound on the first interval. All the other numbers can be junked
Hope that is of help even if it was wordy
identifying values in the domain f(x)=x-3/x+2
The rational function f(x) = (x - 3) / (x + 2) have a domain corresponding to any real number except x = - 2.
What is the domain of a rational equation?
In this problem we find a defined rational functions whose numerator and denominator are both linear equation. The domain of a function is the set of all values of x such that f(x) exists.
By function theory, the domain of all linear equations is the set of all real numbers and the domain of rational equations is the domain of the numerator except the values of x such that the denominator becomes zero.
Then, we need to determine the values of x such that:
x + 2 = 0
x = - 2
The domain of the rational function f(x) = (x - 3) / (x + 2) is all the real numbers except x = - 2.
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Find the y-intercept of the line 12x+16y=6.
Answer:
[tex]\frac{3}{8}[/tex]
Step-by-step explanation:
To find the y-intercept you need to put convert it into the equation:
y = mx + b where b = the y-intercept.
12x + 16y = 6. First rearrange the equation so y is on side of the equation.
16y = -12x + 6. Now divide by 16 so the equation has y as a stand alone value.
y = [tex]\frac{-3}{4}[/tex]x + [tex]\frac{3}{8}[/tex]
Therefore the y-intercept is [tex]\frac{3}{8}[/tex]
Consider the following data of mathematics students.
60 study French, 30 study French and German
55 study German, 25 study French and Russian
75 study Russian, 35 study German and Russian
12 students study all the three languages.
Let F, G, R denote the set of students studying French, German and Russian respectively. Then find the number of students studying at least one of the three languages i.e, n(FUGUR)
The value of n (FUGUR) will be;
⇒ n (FUGUR) = 112
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The data of mathematics students is,
60 study French, 30 study French and German
55 study German, 25 study French and Russian
75 study Russian, 35 study German and Russian
12 students study all the three languages.
Now,
Let F, G, R denote the set of students studying French, German and Russian respectively.
And, The data of mathematics students is,
n (F) = 60, n (F ∩ G) = 30
n (G) = 55, n (F ∩ R) = 25
n (R) = 75 , n (G ∩ R) = 35
n (F ∩ G ∩ R) = 12
Since, We know that;
⇒ n (FUGUR) = n (F) + n (G) + n (R) - n (F ∩ G) - n (F ∩ R) - n (G ∩ R)
+ n (F ∩ G ∩ R)
Substitute all the values, we get;
⇒ n (FUGUR) = n (F) + n (G) + n (R) - n (F ∩ G) - n (F ∩ R) - n (G ∩ R)
+ n (F ∩ G ∩ R)
⇒ n (FUGUR) = 60 + 55 + 75 - 30 - 25 - 35 + 12
⇒ n (FUGUR) = 112
Thus, The value of n (FUGUR) will be;
⇒ n (FUGUR) = 112
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