A vertex is a point where two or more curves, lines, or edges meet.
In geometry, vertex generally refers to an intersection point or a corner where lines, curves or edges meet. As a consequence of this definition, the point where two lines meet to form an angle called an included angle are considered vertices. Hence, the corners of triangle, polygons and polyhedral are vertices. For example, the lines of square or rectangle intersect at four corners which are called its vertices (which is the plural of vertex). Similarly, the vertex of curve(s) is where they intersect with each other or with its line of symmetry.
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What is the statement of triangle?
The statement of triangle is that it has three corresponding sides and three corresponding angles.
What are the theorems about triangles?There are at least 3 theorems about triangles, those are: the total of the three interior angles in any triangle is 180 degrees; when a triangle side is constructed, the exterior angle formed is equal to the sum of the interior opposite angles; and, the base angles of an isosceles triangle are equivalent.
Scalene, isosceles and equilateral triangle are the types of triangles which differ from each other based on their side-length. If all the three sides are different in length, then its scalene triangle. If any two sides are equal in length, then it is an isosceles triangle.
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which is the equation of the line with a slope of 2 that goes through the point (1,2.5)
Answer:
y - 2.5 = 2(x - 1)
Step-by-step explanation:
Here given slope is 2 and the point through which the straight line passes is (1, 2.5)
Let the equation be y = mx + c where m and c represent slope and y intercept respectively.
We know the slope, so we input that in our equation
y = 2x + c ------- i
This straight line also passes through (1, 2.5)
Putting the values in place of x and y we get,
2.5 = (2X1) + c
⇒ 2.5 = 2 + c
⇒ c = 2.5 - 2
⇒ c = 0.5
Now, putting value of c in i, we get
y = 2x + 0.5 -------- ii
It can also be written as y - 2.5 = 2(x - 1) ------- iii
this equation and ii are same and you can verify that by simplifying iii, by multiplying 2 and then shifting the -2.5 from LHS to RHS
Graph the line whose y intercept is -9 and whose x intercept is 7
The graph of the linear equation y = 9/7x - 9 is attached below
Graph of a Linear EquationA linear graph is the graphical representation of a straight line. It showcases a linear equation in two variables and shows the linear association between two quantities. A linear function often have a straight line with a slope and y-intercept.
To solve this, we can simply write an equation of the linear equation which is often given as
y = mx + c
m = slopec = y-interceptLet's substitute the values and find the equation;
y = 9/7x - 9
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A line has a slope of 1 and passes through the point (8, –1) . What is its equation in slope -intercept form?
The slope-intercept form of a linear equation is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept (the point at which the line crosses the y-axis).
Given that the slope of the line is 1 and it passes through the point (8, –1), we can use the point-slope form of a linear equation to write the equation in slope-intercept form. The point-slope form is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Substituting the values given in the problem into the point-slope form, we get:
y - (-1) = 1(x - 8)
Expanding the right side and rearranging the terms, we get:
y + 1 = x - 8
y = x - 9
This is the equation of the line in slope-intercept form. The y-intercept is (-9, 0), which is the point at which the line crosses the y-axis.
What is 1/2 equivalent to as a fraction?; Why are 1/2 and 2/4 called equivalent fractions?; What is 1/3 equivalent to as a fraction?; How do I find the equivalent fraction?
The equivalent fraction of 1/2 is 2/4 and equivalent fraction of 1/3 is 2/6.
Given that,
Why are the fractions 1/2 and 2/4 referred to as identical, and what fractional value does 1/2 equal? What is 1/3's equivalent in fractions
The fractions 2/4, 3/6, 4/8, etc. are identical to 1/2. Reduced fractions of equivalent fractions have the same value. Simply multiplying or dividing both the numerator and the denominator by the same number produces equivalent fractions.
By multiplying the numerator and denominator of each fraction by the same number, we can determine the equivalent fraction for each fraction. For instance, we must multiply 2/3 by 3/3 after determining the third equivalent fraction of 23. Consequently, 2/3 (3/3) = 6/9 is the fraction that equals 2/3.
As a result, the equivalent fractions of 1/2 and 1/3 are 2/4 and 2/6, respectively.
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The total cost of the snow cones, t, depends on the number of snow cones ordered, s. The relationship is shown in the table.
Number of
Snow Cones Total Cost
(dollars)
2 6
4 12
5 15
Which equation can be used to calculate the total cost, t, of s snow cones?
The equation will be t = 3s where t is total cost and s is number of snow cones. this can be find out be relationship of number of snow cones and total cost in table .
In given table :
Number of snow cones Total cost
2 6
6 12
5 15
From above table, we can relate
1st case :
number of snow cones = 2 and total cost = 6
i.e. 3 times of number of cones.
2nd case:
number of cones = 4 and total cost = 12
i.e. 3 times the number of cones.
3rd case :
number of cones = 5 and total cost = 15
i.e. 3 times the number of cones.
So, we see same pattern is repeated in all.
Then equation will be t = 3s where t = total cost , s = number of snow cones.
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The SC Electric Company has bid on two electrical wiring jobs. The owner of the company believes that The probability of being awarded the first job (event A) is 0.75; . The probability of being awarded the second job (event B) is 0.5; and The probability of being awarded both jobs (event (A and B)) is 0.375.
If the owner's beliefs are correct, Event A and Event B are not mutually exclusive and are independent and must be true for event A and event B.
Simply put, the probability is the likelihood that something will occur. When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes.
By simply dividing the favorable number of possibilities by the entire number of possible outcomes, the probability of an occurrence can be determined using the probability formula. Because the favorable number of outcomes can never exceed the entire number of outcomes, the chance of an event occurring might range from 0 to 1. Additionally, the proportion of positive outcomes cannot be negative. In the sections that follow, let's go into greater detail on the fundamentals of probability.
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assume that for each n, fn is an integrable function on [a, b]. further, assume that the sequence (fn) converges uniformly to f on [a, b]. prove that f is integrable on [a, b].
fn is an integrable function on [a, b]. further, assume that the sequence (fn) converges uniformly to f on [a, b]. f is integrable on integral [a, b].
Let ε > 0. Since (fn) converges uniformly to f on [a, b], there exists an N such that |fn(x) - f(x)| < ε for all x ∈ [a, b] and all n ≥ N.
Since fn is integrable on [a, b], it follows that
∫a b|fn(x)|dx < ∞ for all n ≥ N.
We have
∫a b|f(x)|dx ≤ ∫a b|fn(x)|dx + ∫a b|fn(x) - f(x)|dx
Since |fn(x) - f(x)| < ε, it follows that
∫a b|f(x)|dx ≤ ∫a b|fn(x)|dx + ε(b - a)
Since the right hand side is finite, it follows that ∫a b|f(x)|dx is finite. Hence, f is integrable on integral [a, b].
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A botanist wishes to estimate the typical number of seeds for a certain fruit. She samples 65 specimens and counts the number of seeds in each. Use her sample results (mean = 77.3, standard deviation = 5.5) to find the 80% confidence interval for the number of seeds for the species. Enter your answer as an open-interval (i.e., parentheses) accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).80% C.I. =Answer should be obtained without any preliminary rounding.
The answer is (76.42 ; 78.17)
Given that:
A botanist is attempting to determine how many seeds are typically present in a particular fruit. She collects 65 samples and counts how many seeds are present in each. Using her sample data, the mean is 77.3 and the SD is 5.5.
Sample size, n = 65
Mean, xbar = 77.3
Standard deviation, s = 5.5
Confidence level, Zcritical at 80% = 1.28
Confidence interval :
Xbar ± Margin of error
Margin of Error = Zcritical * s/sqrt(n)
Margin of Error = 1.28 * 5.5/sqrt(65)
Margin of Error =0.873
Lower boundary = 77.3 - 0.873 =76.42
Upper boundary = 77.3 + 0.873 = 78.17
Therefore, the answer is (76.42 ; 78.17)
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Suppose you are trying to learn the relationship between the price you charge for your product and the likelihood of purchase by individuals offered that price. You offer your product online and for one month have randomly posted prices between $10 and $30. Using data on purchases and prices, you get the following estimates for a linear probability model:
Purchasei=1.7−0.06×Pricei
You are interested in the effect of a $20 price increase (i.e., moving from the lowest price to the highest) on the likelihood of Purchase. Why is answering this question problematic using this model?
The issue with this model is that purchases with a cost of $30 appear to be negative, which is impractical in real-world situations.
This is due to the model's linear best fit line's potential for being inaccurate for the end points.
A linear probability model:
Purchasei = 1.7-0.06×pricei
For an increase in $20 in price,
Change in purchase = -0.06 × 20
change in purchase = -1.2
There would be 1.2 less purchase.
Purchase at price of $10 is:
Purchase 10= 1.7 ×0.06 × 10
Purchase 10 = 1.1
Purchase at price of $30 is:
Purchase 30 = 1.7 × 0.06 × 30
Purchase 30 = -0.1
This is the problem with this model that the purchases at price of $30 is coming as negative which is not possible in practical situation.
It is because that the linear best fit line of the model which might not be accurate for the end points.
Hence we get the required estimates here:
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Change the circle equation into standard form then answer the following questions:
Coordinates of the center?
Coordinates of the vertices?
How long is the radius?
Domain?
Range?
X²+ Y²-6X-2Y+6=0
The circle equation in standard form is (X - 3)² + (Y - 1)² = 2²
How to change the circle equation into standard form?
We must transform the current polynomial to circle standard form:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
We will do so by completing the square, which produces a perfect square trinomial that can be factored into a square binomial in the form of the circle standard form.
According to the given question:
Equation of circle in polynomial form is X²+ Y²- 6X - 2Y + 6 = 0
Converting to standard form using completing the square method
X²+ Y²- 6X - 2Y + 6 = 0
(X² - 6X) + (Y² - 2Y) = -6
(X² - 6X + 9) + (Y² - 2Y + 1) = -6 + 9 + 1
(X - 3)² + (Y - 1)² = 4
(X - 3)² + (Y - 1)² = 2²
The circle is now in standard form.
Coordinates of the center (h,k) = (3, 1)
Coordinates of the vertices = (3, 1), (3, -6), (2, -4), (8, -1)
Domain = [2, 8]
Range = [-6, 1]
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[-1(4x-2) -5] [-1 2 -5]
[0 (3z-9) (y-1)] [0 3 2] value y?
The value of y is 4.
What is The value of y?Generally, To find the value of y in the expression [0 (3z-9) (y-1)], we need to isolate y. To do this, we can start by combining like terms:
[-1(4x-2) -5] [-1 2 -5] [0 (3z-9) (y-1)] [0 3 2]
Becomes:
[-4x + 2 - 5] [0 3 2]
Then we can rearrange the terms to get:
[-4x - 5 + 2] [0 2 + 3]
This simplifies to:
[-4x - 3] [5]
Finally, we can isolate y by dividing both sides by 2:
[-4x - 3] / 2 = 5 / 2
This gives us:
y = (5/2) + 3/2
Simplifying, we find that y = 4.
Therefore, the value of y in the expression [0 (3z-9) (y-1)] is 4.
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Determine between which consecutive integers the real zeros of f(x)=x²-4x-2 are located.
a. between 3&4 and -1&0
c. between 4&5 and -1&0
between 3&4 and 0&1
b. between 4&5 and 0&1
d.
Please select the best answer from the choices provided
Answer:
c. between 4&5 and -1&0
Step-by-step explanation:
Is this a linear function? y-8= -8(x-2)
Answer:
Yes
Step-by-step explanation:
y-8= -8(x-2) ==> solve for y
y = -8(x - 2) + 8 ==> add 8 on both sides to isolate y
y = (-8 * x - 2(-8)) + 8 ==> distribute -8 to x and -2 using the distribution
property
y = (-8x - (-16)) + 8 ==> simplify
y = (-8x + 16) + 8 ==> subtracting by a negative number is equivalent to
adding by a positive number.
y = -8x + 16 + 8
y = -8x + 24 ==> Degree of x in the equation is 1, so the function is linear
20,245,70 Find the LCM of the following whole numbers.
Answer: 980
Step-by-step explanation: 20 = 2 x 2 x 5
245 = 5 x 7 x 7
70 = 2 x 5 x 7
= 2 x 2 x 5 x 7 x 7
= 980.
Greatest common divisor: 5
the difference of set1 and set2 is a set that contains only the elements that appear in set1 but do not appear in set2. True/False
The difference operation in set removes every common element between the set and give the remaining element of first set. The answer is True.
What is a set?In mathematics, a set is a logically arranged group of items that can be represented in either set builder or roster form.
What is Set Difference?The set that contains the items of S that are not elements of T is the difference between the two sets, denoted by the symbol S-T.
set1={1,2,3,4,5}
set2={4,5,6,7}
Set Difference= set1-set2 = {1,2,3}
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Evaluate for d = 10, u = 7, e = 3.2 d+e
Answer:
The answer is 13.2
Step-by-step explanation:
d = 10
e = 3.2
d + e = 10 + 3.2 = 13.2
Thus, The answer is 13.2
what are the values of x and the measure of angle e to the nearest degree?
The value of x or we can say that the value of side DF will be equal to 5.29.
What is an equation?Mathematical expressions with two algebraic symbols on either side of the equal (=) sign are called equations.
This relationship is illustrated by the left and right expressions being equal. The left-hand side equals the right-hand side is a basic, straightforward equation.
As per the given information in the question,
The sides of the right triangle are,
DE, h = 8
FE, b = 6
DF, p = x
Then, use Pythagoras' theorem,
h² = x² + b²
8² = x² + 6²
x² = 64 - 36
x = √28
x = 5.29
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assume that the salaries of fast food restaurant managers are normally distributed with a mean of $48,000 and a standard deviation of $4,000. what is the lowest amount that a manager needs to make and still be one of the top 7% earners? use excel, and round your answer to the nearest dollar. provide your answer below:
According to the provided data, to be one of the top 7% earners a manager needs to earn $48,360.
What do you mean by statistics?The study of statistics focuses on gathering, organizing, organizing, analyzing, interpreting, and presenting data.
What is the formula to calculate mean and standard deviation?The formulas are given below:
[tex]\mu = \frac {\sum x}{N}[/tex] (Mean)
[tex]\sigma = \sqrt {\frac {\sum (X- \mu ) ^ {2}} {N}[/tex] (Standard Deviation)
Mean = $48,000
Standard Deviation = $4,000
According to this data, Max Earning made = $48,000 + $4,000
=$52,000
7% of $52,000 = $3,640
To be top 7% earner, A manager must earn = $52,000 - $3,640
=$48,360
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Answer:$53,903
Step-by-step explanation:
Here, the mean, μ, is 48,000 and the standard deviation, σ, is 4,000. Let x be the cutoff salary for teachers in the top 7%. The area to the right of x is 7%=0.07. So, the area to the left of x is 1−0.07=0.93.
Open Excel. Click on an empty cell. Type =NORM.INV(0.93,48000,4000) and press ENTER.
Round the answer to the nearest dollar, is x≈53,903. Therefore, the cutoff salary for managers in the top 7% is $53,903.
Why did the duck fly south for the winter
Answer:
warmth
Step-by-step explanation:
they migrate to stay warm I hope you do good (:
Answer:
To find a warmer place for nesting grounds or migration.
Step-by-step explanation:
The Fitzhugh-Nagumo model for the electrical impulse in a neuron states that, in the absence of relaxation effects, the electrical potential in a neuron v(t) obeys the differential equation dv dt = −v[v2 − (1 + a)v + a] where a is a positive constant such that 0 < a < 1. (a) For what values of v is v unchanging (that is, dv/dt = 0)? (Enter your answers as a comma-separated list.) v = (b) For what values of v is v increasing? (Enter your answer using interval notation.) (c) For what values of v is v decreasing? (Enter your answer using interval notation.)
The values of v for the electric impulse model in the Fitzhugh-Nagumo model are 0, 1, and a.
What is electric impulse?The strength of the field is shown by the distance between the lines; the closer they are to one another, the stronger the field.
In this scenario, the field weakens as we travel away from the charge because the distance between the lines widens (it actually obeys an inverse square law,
The electric field, which is pointed away from the core charge, is indicated by the direction of the lines.
This is due to the fact that a positive test charge would be repelled away from the electric field if it were placed there because the direction of the electric field corresponds to the direction of the force that a positive test charge would experience when immersed in the electric field.
Hence, The values of v for the electric impulse model in the Fitzhugh-Nagumo model are 0, 1, and a.
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The equation s equals two fifths times f represents the proportional relationship between the amount of sugar (s) and flour (f) in a recipe. Which table represents the equation of the proportional relationship?
Sugar Flour
1 two fifths
2 four fifths
Sugar Flour
four fifths 1
two fifths 2
Sugar Flour
1 four fifths
2 two fifths
Sugar Flour
two fifths 1
four fifths 2
The table which represents the equation of the proportional relationship as described is;
Sugar Flour
two fifths 1
four fifths 2
Which table represents the equation of the proportional relationship?It follows from the task content that the table which represents the proportional relationship is to be identified among the given answer choices.
Since the given relationship is defined by the equation;
s = (2/5) f.
Hence, when f = 1; s = (2/5) • 1; s = 2/5.
Also, when f = 2; s = (2/5) • 2 = 4 / 5.
Therefore, the table which represents the equation of the proportional relationship is;
Sugar Flour
two fifths 1
four fifths 2
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The graph of the function f(x) = (x – 4)(x + 1) is shown below.
On a coordinate plane, a parabola opens up. It goes through (negative 1, 0), has a vertex at (1.75, negative 6.2), and goes through (4, 0).
Which statement about the function is true?
The function is increasing for all real values of x where
x < 0.
The function is increasing for all real values of x where
x < –1 and where x > 4.
The function is decreasing for all real values of x where
–1 < x < 4.
The function is decreasing for all real values of x where
x < 1.5.
The function is increasing for all real values of x where
x < –1 and where x > 4.
What is increasing and decreasing functions?
Calculus uses a function's derivative to determine whether a function is increasing or decreasing at various intervals in a specific domain. For a given function, y = F(x), the function is known as an increasing function if the value of y increases as the value of x increases, and the function is known as a decreasing function if the value of y decreases as the value of x increases.
Given function is
f(x) = (x - 4)(x + 1)
If f(x) = 0, x=4 & x=-1
f'(x) = 0, x=3/2
So, we will check the behavior of the function in the neighborhood of x=4,-1, 3/2.
What is the increasing and decreasing function?
A function is said to be an increasing function if its slope is continuously increasing in a given interval.
A function is said to be a decreasing function if its slope is continuously decreasing in a given interval.
If x>4
Let us check at x=5
f(5) =6(+ve)
f(x) >0 for x>4
So, the function is increasing in x>4
Similarly, If x <-1
f(x) >0 for x <-1
So, function is increasing in x <-1
If -1<x<3/2
f(x)<0 for -1<x<3/2
So, function is decreasing in -1<x<4
If 3/2<x<4
f(x)>0 for 3/2<x<4
So, the function is increasing in 3/2<x<4
From the graph too, we can see the behavior of the given function
by observing the slope.
We can see that for x<-1 and x>4, the slope is continuously increasing
So, the function is increasing in x<-1 and x>4.
Therefore, the function is increasing for all real values of x where
x < –1 and where x > 4.
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Tell whether the table of values represents a linear, exponential, or quadratic function.
The table of values represents a quadratic function. And a quadratic function is y = x² / 10.
What is a quadratic function?
A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function.
Linear functions have constant first differences.
Here, if a and b is the first and second term.
Then the first difference is b-a.
Quadratic functions have constant second differences.
Let the first differences are:
b-a = m, c-b = n
Then the second difference is n-m.
Exponential functions have a constant ratio.
The ratio of first term and second term is b/a.
Here, from the table;
y₂-y₁ = 0.4 - 0.9 = -0.5 = m₁
y₃-y₂ = 0.1 - 0.4 = -0.3 = m₂
y₄-y₃ = 0 - 0.1 = -0.1 = m₃
y₅-y₄ = 0.1 - 0 = 0.1 = m₄
y₆-y₅ = 0.4 - 0.1 = 0.3 = m₅
From the first differences, we have no idea about the function property.
So, the second differences:
m₂ - m₁ = -0.3 + 0.5 = 0.2
m₃ - m₂ = -0.1 + 0.3 = 0.2
m₄ - m₃ = 0.1 + 0.1 = 0.2
m₅ - m₄ = 0.3 - 0.1 = 0.2
Here, the second differences are constant 0.2.
And the common ratio is varying.
So, quadratic functions have constant second differences.
And the quadratic function is y = x² / 10.
Therefore, the given function is a quadratic function.
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Control refers to
A. directly manipulating an independent variable in a research study.
B. managing unwanted variables that could influence the results of a research project.
C. Both (a) and (b).
D. None of the above.
Correct option is A
Independent variable
Independent variable is manipulated by the researcher to determine the effect of its change relative to a change in dependent variable.
Independent Variables
An independent variable is exactly what it sounds like. It is a variable that stands alone and isn't changed by the other variables you are trying to measure. For example, someone's age might be an independent variable.
In an experiment the independent variable is the variable that is varied or manipulated by the researcher. The dependent variable is the response that is measured. One way to think about it is that the dependent variable depends on the change in the independent variable.
Qualitative research consist in observing some variables and obtaining information. Usually the qualitative research does not involve the manipulation of independent variables, these types of researches focuses more in collecting answers in groups of people, and making statistics with those answers or information. So the variables are not manipulated.
Correct option is A
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Find the value of √17+4√13-√17-4√13
Answer:0
Step-by-step explanation:
By adding like terms,
4√13 and -4√13 equal to 0, and √17 and -√17 equal 0
is the following true? pls i need help
Answer: No, i dont think so.
Step-by-step explanation:
AC = BC is not listed as given.
A construction worker earns and average biweekly net pay of $1,764. Which compound inequality correctly shows the amount of money, t, the worker can spend if the monthly budget for food is between 10% and 20% ?
The compound inequality of amount of money is 176.4 < f < 352.8
What is compound inequality?
A compound difference (or combined difference ) is 2 or a lot of inequalities joined along with or or and.
Main Body:
Given,
A construction worker earns an average bi-weekly net pay of $1,764.00.
and, the worker can spend if the monthly budget for food is between 10% and 20%.
To find the which compound inequality correctly shows the amount of money?
Now, According to the question:
Total earning is $1,764.00
Amount can be spent on food is :
0.1× 1764.00 < f < 0.2 ×1764.00
176.4 < f < 352.8
Hence, The compound inequality of amount of money is 176.4 < f < 352.8
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1. Mark says the two expressions (9a-27)-4
and -3 (-4-3) are equivalent is he correct
Explain how you know.
They (circle one) are/arenot
equivalent because
Show your work
Equations are not equivalent because the values are found to be different.
What is an expression?
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.
Expressions are:
(9a - 27) - 4 = 9a - 31 -------(I)
and -3(- 4 - 3) = -3(-7) = 21 -----(II)
From (I) and (II), expressions are not equivalent.
Hence, equations are not equivalent because the values are found to be different.
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For the functions f(t) = et and g(t) = e-6t, defined on 0 Le t Le infinity, compute f * g in two different ways: By directly evaluating the integral in the definition of f * g. (f * g) (t) = dw = By computing L-1 {F(s)G(s)} where F(s) = L{f(t)} and G(s) = L{g(t)}. (f * g)(t) = L-1 {F(s)G(s)} = L-1{ }
The value of the expression (f * g) by using two ways that is by direct integral and by Laplace transform is equal to (1/2)e^t/(1-t) - (1/2)e^t/(1+t) + C.
To compute the convolution of f(t) and g(t) by directly evaluating the integral in the definition of the convolution, we have:
(f * g)(t) = ∫f(w)g(t-w)dw
Substituting in the expressions for f(t) and g(t), we have:
(f * g)(t) = ∫e^w*e^(-6)(t-w)dw
This integral can be evaluated as follows:
(f * g)(t) = ∫e^(-5w)dw
(f * g)(t) = (-1/5)e^(-5w) + C
(f * g)(t) = (-1/5)e^(-5w) + C
To compute the convolution of f(t) and g(t) by computing the inverse Laplace transform of the product of the Laplace transforms of f(t) and g(t), we have:
F(s) = L{f(t)} = ∫f(t)e^(-st)dt = ∫e^t*e^(-st)dt = 1/(s-1)
G(s) = L{g(t)} = ∫g(t)e^(-st)dt = ∫e^(-6t)*e^(-st)dt = 1/(s+6)
Therefore,
F(s)G(s) = (1/(s-1))(1/(s+6)) = 1/(s^2-1)
To find the inverse Laplace transform of F(s)G(s), we use the following formula:
L^-1{F(s)G(s)} = ∫F(s)G(s)e^st ds
Substituting in the expression for F(s)G(s), we have:
L^-1{F(s)G(s)} = ∫(1/(s^2-1))e^st ds
This integral can be evaluated as follows:
L^-1{F(s)G(s)} = (1/2)e^st/(s-1) - (1/2)e^st/(s+1) + C
L^-1{F(s)G(s)} = (1/2)e^st/(s-1) - (1/2)e^st/(s+1) + C
Thus, the convolution of f(t) and g(t) can be computed either by directly evaluating the integral in the definition of the convolution or by computing the inverse Laplace transform of the product of the Laplace transforms of f(t) and g(t). In both cases, the result is:
(f * g)(t) = (1/2)e^t/(1-t) - (1/2)e^t/(1+t) + C
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