Answer: 6 square root 23
Step-by-step explanation:
10^3 + 8^2
How do you find a discriminant for a 2x2 matrix?
The discriminant of a 2x2 matrix is equal to the determinant of the matrix squared. The determinant is calculated by subtracting the product of the two diagonal elements from the product of the two off-diagonal elements
To find the discriminant of a 2x2 matrix, first calculate the determinant of the matrix. The discriminant is then equal to the determinant of the matrix squared.
For example, consider the 2x2 matrix:
A = [a b]
[c d]
The determinant of A is equal to (ad - bc). The discriminant of A is then equal to (ad - bc)^2.
The discriminant of a 2x2 matrix is equal to the determinant of the matrix squared. The determinant is calculated by subtracting the product of the two diagonal elements from the product of the two off-diagonal elements.
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Do the one you would like please show work
6) The time required by James if he completes the work alone is 18 hours.
What is time and work?Time and work are concerned with the amount of time it takes a person or a group of people to complete a task and the effectiveness of the work performed by each of them. The fundamental idea behind Time and Work is comparable to the idea behind Proportionality, which is present in all arithmetic disciplines.
"How much work one person can complete in one day is what the term "work efficiency" refers to. For instance, someone can complete a task in two days. In other words, he may complete 50% of the work in a single day. His effectiveness will therefore be 50%.
Given that,
Monica is fencing a yard
Monica took 9 hours to fence the yard.
If she takes James help to fence the yard then she estimates that she can do it in 6 hours.
Now, we have to find how much time James requires if he work alone=?
Then,
(9x)/(9+x)=6
9x=54+6x
3x=54
x=18
So, the time required by James if he completes the work alone is 18 hours.
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How does the graph of y √ x 2 compare to the graph of the parent square root function?
Answer:
to move te function up z units, add zto the whole equation
from
y=sqrt(x) to
y=sqrt(x)+2
parenth function moved up 2 units
my hands are cold so I don't type well
Step-by-step explanation:
How do you calculate slope quickly?
The process to calculate slope of line is explained below.
The Slope of line passing through two points (a,b) and (c,d) can be calculated by using formula ;
slope = (d - b)/(c - a) ;
The slope of a line is denoted by letter "m" .
For Example : If the line is passing through the points (0,2) and (3,8) .
on comparing we get the values as a = 0 , b = 2 , c = 3 and d = 8 .
On substituting values in the slope formula that is mentioned above ,
we get ;
Slope(m) = (8 - 2)/(3 - 0)
On Simplifying further ,
we get ;
= 6/3
= 2 .
So , the slope of line will be = 2 .
The given question is incomplete , the complete question is
How do you calculate slope of a line quickly ?
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x+12)^(2)=x^(2)+24x+ __ what is the missing term?
To find the missing term, we can use the expansion of the square of a binomial, which is given by the formula (a + b)^2 = a^2 + 2ab + b^2. In this case, the binomial is x + 12, so we can use the formula like this:
[tex](x + 12)^2 = x^2 + 2 * x * 12 + 12^2[/tex]
Therefore, the missing term is 144.
You can also find the missing term by expanding the left-hand side of the equation using the same formula:
[tex](x + 12)^2 = (x^2 + 24x) + __[/tex]
When we expand the left-hand side using the formula, we get:
[tex](x + 12)^2 = x^2 + 24x + 144[/tex]
So the missing term is also 144.
What is the inverse of 7 mod 11?
The inverse of 7 modulo 11 is 4, which means that 4 multiplied by 7 gives a number congruent to 1 (mod 11). This means that 4 is the number that when multiplied by 7 gives a remainder of 1 when divided by 11.
In modular arithmetic, the inverse of a number is the number that when multiplied by the original number gives a result that is congruent to 1 (modulus). For example, if we take the modulus of 11 as an example, then the inverse of 7 modulo 11 is 4, since 7*4 = 28 ≡ 1 (mod 11). This means that 4 is the number that when multiplied by 7 gives a remainder of 1 when divided by 11. The inverse of a number modulo a given modulus can be calculated using the Extended Euclidean algorithm. This algorithm allows us to find the inverse of a number modulo a given modulus by solving a series of equations and using the result to calculate the inverse. For example, in the case of 7 mod 11, the Extended Euclidean algorithm gives us the result 4, which is the inverse of 7 mod 11.
11 = 7*1 + 4
7 = 4*1 + 3
4 = 3*1 + 1
3 = 1*3 + 0
By reading the equations from right to left, we can see that the inverse of 7 mod 11 is 4, since 4*7 = 28 ≡ 1 (mod 11).
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What is the median of 5 and 7?
The median of 5 and 7 is 6.
The median is the middle point in a dataset—half of the data points are smaller than the median and half of the data points are larger.
A data set's median value is the point where 50% of the data points have values that are lower or equal to it, and 50% of the data points have values that are higher or equal to it.
To arrange the data points in ascending order for a small data set, count the number of data points (n).
To get the rank of the data point whose value is the median when the number of data points is unequal, multiply the total by 1 and divide the results by 2.
Median = (5+7)/2 = 12/2 = 6
Thus, the median of 5 and 7 is 6.
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To get the rank of the data point whose value is the median when the number of data points is unequal, multiply the total by 1 and divide the results by 2.
Median = (5+7)/2 = 12/2 = 6
Thus, the median of 5 and 7 is 6.
How many numbers are divisible by 3 from 100 to 1000?
On solving the provided question, we can say that - There exist 333 integers smaller than 1000 that are evenly divisible by 3 if you divide 1000 by 3, which results in 333 with a residual of 1.
what is divisibility test?The divisibility rule is a concise and practical approach to check, often by looking at the integer's digits, whether a given integer is divisible by a given set divisor without actually executing division. Without actually doing the division procedure, you may quickly discover if a given number can be divided by a defined divisor using the divisibility test. When dividing two numbers exactly, the quotient must be an integer and the remainder must be zero.
here,
1000/3
remainder = 1
divisor = 333
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H(x)= 3x^4+2x—5/x+9x^3-7, is it a polynomial? If so write in standard form, degree, type, and leading coefficient.
The function H(x) = 3x⁴ + 2x — 5/x + 9x³ - 7 is not a polynomial function
What is a polynomial function?A polynomial function is a function in an equation, such as the quadratic equation, cubic equation, etc., that only uses non-negative integer powers or only positive integer exponents of a variable.
A polynomial with an exponent of 1 is, for instance, 2x+5.
The given polynomial in standard form is written as
H(x) = 3x⁴ + 2x — 5/x + 9x³ - 7
H(x) = 3x⁴ + 9x³ + 2x - 7 - 5x⁻¹
this shows that on of the term has a negative exponent 5x⁻¹ and hence not a polynomial
the leading coefficient is 3
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By accident, janae has been making decaffeinated coffee for her co-workers for the entire week. But some of her co-workers have claimed that the extra boost of caffeine has helped them focus on their work. What does the placebo effect mean in this specific situation?
the placebo effect means in this specific situation that The benefit is due to the fact that the co-workers are given a "treatment" (coffee, although decaffeinated) and not an effect of the "active ingredient" (caffeine).
Therefore, it appears that Janae is serving decapitated coffee to its customers, who have said that it has given them more energy. We want to examine the placebo effect, and when we think of the placebo effect, we imagine that a guy should publish. The best response you have provided is to recognize that they are receiving treatment, even though it is not caffeine, and causing them to believe that they are experiencing an increase in energy and that it is not the result of an active ingredient. As a result, they are receiving treatment and are having an impact by which. They aren't actually being impacted by the caffeine, though. You will typically feel more energized as a result of that.
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What is the mode of 1 1 2 2 3 3 4 4?
Answer:
In the set {1, 1, 2, 2, 3, 3, 4, 4}, the value that occurs most frequently is 1, 2, 3, and 4, so the mode of this set is 1, 2, 3, and 4...
The mode of a set of numbers is the value that occurs most frequently in the set. In the set {1, 1, 2, 2, 3, 3, 4, 4}, the value that occurs most frequently is 1, 2, 3, and 4, so the mode of this set is 1, 2, 3, and 4.
It's important to note that a set of numbers can have more than one mode if multiple values occur with the highest frequency. A set of numbers that has multiple modes is called a multimodal set. In the example set {1, 1, 2, 2, 3, 3, 4, 4}, all four values occur with the same frequency, so the set is multimodal.
On the other hand, if no value occurs more frequently than any other value, the set is said to have no mode. For example, the set {1, 2, 3, 4, 5} has no mode because all values occur with the same frequency.
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Kelly is making new throw pillows for her couch and wants to make 5 new pillows. The
pillows will be different types of material-2 pillows made of linen, 2 out of satin, and 1 out of faux fur. Each pillow needs 2 yards of fabric. The linen fabric costs $13.12 per yard, the satin fabric costs $9.99 per yard, and the faux fur costs $14.99 per yard. She also needs to buy the pillow inserts, which are $9.99 each. What is the total cost Kelly spends on the materials for the pillows?
Answer:
The total cost of the materials for the pillows is $172.37
Step-by-step explanation:
First, we need to calculate how many yards of each type of fabric Kelly needs to buy. She needs 2 linen pillows, which means she needs 22 = 4 yards of linen fabric.
She also needs 2 satin pillows, which means she needs 22 = 4 yards of satin fabric.
She also needs 1 faux fur pillow, which means she needs 12 = 2 yards of faux fur fabric.
The total cost of the linen fabric is 4*$13.12 = $52.48.
The total cost of the satin fabric is 4*$9.99 = $39.96.
The total cost of the faux fur fabric is 2*$14.99 = $29.98.
The total cost of the materials for the pillows is $52.48 + $39.96 + $29.98 = $122.42.
Finally, Kelly needs to buy 5 pillow inserts at $9.99 each, which means she needs to spend 5*$9.99 = $49.95 on pillow inserts.
The total cost of the materials for the pillows is $122.42 + $49.95 = $172.37.
Does the triangle inequality theorem apply to all equilateral triangles?
a) Find the maximum value of f(x,y) = x^{3}y^{4} for x,y \geq 0 on the unit circle.
The maximum value of f(x,y) on the unit of circle is [tex]\frac{48 \sqrt{21}}{2,401}[/tex].
The full question is in the attachment. From the unit circle
x² + y² = 1y² = 1 - x²Substitute to f(x,y)
f(x,y) = x³y⁴ = x³ (y²)² = x³ (1 - x²)²
f(x,y) = x³ (1 - 2x² + x⁴)
f(x,y) = x³ - 2x⁵ + x⁷
df/dx = 3x² - (2 × 5)x⁴ + 7x⁶
df/dx = 3x² - 10x⁴ + 7x⁶
df/dx = x² (3 - 10x² + 7x⁴)
To find the maximum or minimum value, the derivated equals zero.
0 = x² (7x⁴ - 10x² + 3)
0 = x² (x - 1) (x + 1) (7x² - 3)
x² = 07x² - 3 = 0
Using ABC formula
a = 7b = 0c = - 3[tex]x_{1,2} \:=\: \frac{-b \pm \sqrt{b^2 \:-\: 4ac}}{2a}[/tex]
[tex]x_{1,2} \:=\: \frac{-0 \pm \sqrt{0^2 \:-\: (4 \times 7 \times - 3)}}{2 \times 7}[/tex]
[tex]x_{1,2} \:=\: \frac{\pm \sqrt{4 \times 21)}}{14}[/tex]
[tex]x_{1,2} \:=\: \frac{\pm 2 \sqrt{21}}{14}[/tex]
[tex]x_{1,2} \:=\: \frac{\pm \sqrt{21}}{7}[/tex]
[tex]x_1 \:=\: \frac{\pm \sqrt{21}}{7}[/tex]The maximum value from x₁.
The minimum value from x₂.
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Question 3 (1 point)
What is the quotient? (3.78x10^6) divided by 12,000
O.315
31.5
315
3.15
Answer:
315
Step-by-step explanation:
multiply (3.78x10^6) first to get 3780000 then divide 3,780,000 by 12,000 to get 315
What is a solution of the inequality 3x 2y 12?
The resultant answer after solving the given inequality by taking the subject x is x < (12 + 2y)/3.
What is inequality?An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way.
The majority of the time, size comparisons between two digits on the number line are made.
So, we have the inequality:
3x - 2y < 12
Now, solve the inequality by taking x as the subject as follows:
3x - 2y < 12
3x < 12 + 2y
3x/3 < 12/3 + 2y/3
x < (12 + 2y)/3
Therefore, the resultant answer after solving the given inequality by taking the subject x is x < (12 + 2y)/3.
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Correct question:
What is the solution to the inequality 3x - 2y < 12?
Solve the equation using any method. Provide a reason for your choice.
1. 0=x² + 2x +7
The solution of the equation is x = -1 ± i√6.
What is solution of an equation?
An assignment of values to the unknown variables that establishes the equality in the equation is referred to as a solution. To put it another way, a solution is a value or set of values (one for each unknown) that, when used to replace the unknowns, cause the equation to equal itself.
Given equation is
x² + 2x + 7 = 0
Compare the above equation with ax² + bx + c = 0:
a = 1, b= 2, c = 7.
Now applying quadratic formula x = [-b ± √(b² - 4ac)]/2a:
x = [-2 ± √(2² - 4×1×7)]/(2×1)
x = [-2 ± √(4 - 28)]/(2×1)
x = [-2 ± √(-24)]/2
x = [-2 ± 2i√6]/2
x = -1 ± i√6
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In ΔQRS, s = 5.2 inches, ∠S=129° and ∠Q=30°. Find the length of r, to the nearest 10th of an inch.
The length of r, to the nearest 10th of an inch, is 2.4 inches.
What is the sin angle theorem?The relationship between the sides and angles of non-right (oblique) triangles is defined by the Law of Sines. Simply put, it states that the ratio of the length of a triangle's side to the sine of the angle opposite that side is the same for all triangle sides and angles.
Given that in ΔQRS, s = 5.2 inches, ∠S=129° and ∠Q=30°. The value of side r will be calculated by using the sin angle theorem.
The value of r is,
r / sin21 = s / sin129
r = 5.2 x ( sin21/sin129)
r = 5.2 x 0.46
r = 2.4 inch
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find the equation of the line, help please
Answer:
y = 1x+-5
Step-by-step explanation:
The slope is going up hill so there for the slope is positive. The y-intercept is -5 bc the line crosses at -5 on the y-axis.
ignore the other language
Answer:
x >= 1/3
Step-by-step explanation:
y = (e^x + 1)/3
x = (e^y + 1)/3 ==> inverse function is when you switch x and y
3*x = 3 * (e^y + 1)/3 ==> solve for y by first multiplying both sides by 3
3x = e^y + 1
3x - 1 = e^y + 1 - 1
3x - 1 = e^y ==> now take natural log on each side to isolate y
ln(3x - 1) = ln(e^y)
y = ln(3x - 1) ==> simplify
[tex]f^{-1}[/tex](x) = ln(3x - 1) ==> plug in [tex]f^{-1}[/tex](x) for y to indicate the inverse function
3x - 1 >= 0 ==> you can't have the log function of a negative number
3x - 1 + 1 >= 0 + 1 ==> add 1 on both sides to isolate x
3x >= 1
3x/3 >= 1/3
x >= 1/3
The base of the cone has a radius of 4.5 inches. The height of the cone is 16 inches.
If the cone shown above is sliced by a plane that passes through the center of the base, and also passes through the top vertex, what is the area of the cross-section of the cone?
According to the given statement the area of the cross section of the cone is 72 square units.
What does the word cone actually mean?a solid produced by rotating one of the legs of a right triangle. A solid that is surrounded by a circle and other closed planar base and indeed the surface created by line segments connecting each point here on base's perimeter to a single vertex is known as a right circle cone; for more information, see the Density Formulas Table.
Briefing:From the given information the height is 16 inches and the radius is 4.5 inches becuase the cone is sliced by a plane that passes through the center of the base, and also passes through the top vertex
So, The radius would be half the base, so the base would be 4.5 * 2 = 9 inches.
The area of the cross section of the cone is found by multiplying the base by the height, then dividing by two.
Area of the cross section of the cone = (9 x 16)/2 = 144
144 / 2 = 72 square inches.
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How many terms are there in the expression 5xy +9yz 3zx 5x 4y?
The number of terms that the expression 5xy 9yz 3zx 5x 4y has is 5 terms.
What is an algebraic expression?
An algebraic expression is a set of numbers and letters that make up an expression that has a meaning, the letters are variables and the numbers are coefficients or independent terms, algebraic expressions can be part of an equation and model mathematical processes.
How to determine terms?
To determine the number of terms in an algebraic expression, we only have to count the spaces (addition and/or subtraction symbols) and add the unit, in this way we will have the number of terms.
5xy 9yz 3zx 5x 4y has 4 spaces
4 + 1 = 5 terms.
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How do you write a quadratic equation in standard form given the sum and product of roots?
The quadratic equation in standard form given the sum and product of roots is
x² - (α + β)x + αβ = 0.
What is a polynomial?Polynomial is an equation written as the sum of terms of the form kx^n.
where k and n are positive integers.
We have,
A polynomial with degree 2 is called a quadratic equation.
Now,
The standard form of a quadratic equation is ax² + bx + c.
Now,
Let the roots of the quadratic equation be α and β.
α + β = -b/a
αβ = c/a
Thus,
The quadratic equation is x² - (α + β)x + αβ = 0.
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Which of the following is a zero of the polynomial x3 6x2 11x 6?
The required value zeros of the given polynomial are given as -1, -2, and -3.
What are the zeros of the polynomial?Zeros of the polynomial function are the values of the dependent variable, when putting that value in the expression, the expressions explicitly give zero.
Here,
Let the given expression is equal to zero,
x³+ 6x²+ 11x+ 6 = 0
Factorization of the above expression gives.
(x + 1)(x + 2)(x+3) = 0
Now, for the zeros,
x + 1 = 0; x +2 = 0; x + 3 = 0
x = -1 ; x = -2 ; x = -3
Thus, the zeros of the given expression's zeros are -1, -2, and -3.
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The correct answer for this is x= -1, x = -2 and x =-3
The FLHS gil's basketball team won x games so far this season. The FLHS boy's basketball team won one more than twice as many games. Altotoguether, they won 226 games. How many games did each team win ?
The girl's basketball team won 75 games and the boy's basketball team won 151 games.
Let's assume the number of games the girl's basketball team won "g," and the number of games the boy's basketball team won "b."
We know that the total number of games won by both teams is 226, so we can set up our first equation as follows:
g + b = 226
We also know that the boy's basketball team won one more than twice as many games as the girl's basketball team. We can represent this as follows:
b = 2g + 1
Now we have two equations and two unknown values, so we can solve for g and b using algebra. Let's substitute the second equation into the first equation to eliminate one of the unknowns:
g + (2g + 1) = 226
This simplifies to:
3g + 1 = 226
Subtracting 1 from both sides gives us:
3g = 225
Dividing both sides by 3 gives us:
g = 75
Now that we know the value of g, we can substitute it back into the second equation to solve for b:
b = 2 x 75 + 1 = 151
Therefore, the girl's basketball team won 75 games and the boy's basketball team won 151 games.
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in the xy plane, the circles with equations (x-3) + y = 11 and (x+2) + y = r are externally tangent what is the value or r ?
let me know if you know man
In the XY plane, the circles with equations (x-3) + y = 11 and (x+2) + y = r are externally tangent. the value of r is -6.
What is tangent?Generally, To find the value of r, we can start by solving for the centers and radii of the two circles.
The equation of the first circle is of the form (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. We can rewrite the equation of the first circle as (x - 3)² + y² = 11², so the center of the circle is (3, 0) and the radius is 11.
The equation of the second circle is also of the form (x - h)₂ + (y - k)₂ = r². We can rewrite the equation of the second circle as (x + 2)² + y² = r², so the center of the circle is (-2, 0) and the radius is r.
Since the two circles are externally tangent, the distance between their centers is equal to the sum of their radii. Therefore, the distance between the centers is equal to 11 + r. We can use the distance formula to find the distance between the centers:
distance = √((h₂ - h₁)^2 + (k₂ - k₁)²)
Plugging in the coordinates of the centers, we get:
distance = √((-2 - 3)^2 + (0 - 0)^2) = √(5^2 + 0^2) = √(25) = 5
Since the distance between the centers is equal to the sum of the radii, we can set up the equation:
5 = 11 + r
Solving for r, we get:
r = 5 - 11 = -6
Therefore, the value of r is -6.
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25 startlongdivisionsymbol 1,658 endlongdivisionsymbol to find the quotient, start by dividing into.
The required quotient would be [tex]66\frac{8}{25}[/tex] or 66.32 which is determined by dividing 1,658 by any divisor 25.
Define Quotient?The result of dividing an integer by any so-called quotient divisor is referred to as the quotient. The divisor appears in the dividend a certain number of times.
Arithmetic operations can also be specified by subtracting, dividing, and multiplying built-in functions. The operator that performs the arithmetic operation is called the arithmetic operator.
Multiplication operation: Multiplies values on either side of the operator
For example [tex]12*2=24[/tex]
Division operation: Divides left-hand operand by right-hand operand
For example [tex]\frac{4}{2} =2[/tex]
Dividend is equal to 1,658
Divisor is equal to 25
Unit place value of 1658 is equal to 8.
25 is not a factor of 1658
Smallest number to 1658 divisible by 25 is 1650
Divide 1658 by 25 we get remainder
Dividend = ( Divisor × quotient) + remainder
1658 = 1650 + 8
1658 = ( 25 × 66 ) + 8
Remainder is equal to 8
Quotient is equal to 66
In decimal form
⇒ 1658 ÷ 25
Apply the Division operation, and we get
⇒ [tex]\frac{1658}{25}[/tex]
⇒ 66.32
⇒ [tex]66\frac{8}{25}[/tex]
Therefore,
The required quotient would be [tex]66\frac{8}{25}[/tex] or 66.32 which is determined by dividing 1,658 by any divisor 25.
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How do you describe the nature of the roots of an equation?
The nature of roots of the quadratic equation ax²+bx+c= 0 can be describe in three ways depending on the value of b²-4ac.
For quadratic equation ax²+bx+c=0 if
Case 1) If b²-4ac>0 then the roots of equation will be real and distinct.
Case 2) If b²-4ac=0 then roots will be real and equal.
Case 3) If b²-4ac<0 then roots will be imaginary and complex number pairs.
also a) if b2-4ac >0 and perfect square then the roots are real,distinct, and rational
b) if b2-4ac>0 and not a perfect square then the roots are real, distinct but irrational.
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What is a?? Please help
The value of x when the quadratic expression t² - 8t + 16 is factorised to (t + a)² is -4.
How to factor a quadratic expression?Quadratic expression is an expression with the variable with the highest power of 2. The quadratic equation in its standard form is ax² + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term.
Therefore, let's factorised the expression t² - 8t + 16 in the form (t + a)². Where a is an integer.
Hence,
t² - 8t + 16
t² - 4t - 4t + 16
t(t - 4) -4(t - 4)
(t - 4)(t - 4)
Hence,
(t + a)² = (t - 4)²
Therefore,
a = -4
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The inverse of the function f (x) = e^x-e^- x/e^x+e^- x + 2 is given by
a.log (x - 2/x - 1 )^1 2
b.12loge (x - 1/3 - x )
c.&log (x/2 - x )^1 2 &
d.log (x - 1/3 - x )^1 2
We're asked to find inverse of the function,
[tex]\longrightarrow\rm{f(x)=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}+2}[/tex]
Take [tex]\rm{y=f(x).}[/tex]
[tex]\longrightarrow\rm{y=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}+2}[/tex]
Subtract 2.
[tex]\longrightarrow\rm{y-2=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}}[/tex]
We apply componendo - dividendo rule.
[tex]\rm{\dfrac{a}{b}=\dfrac{c}{d}\quad\iff\quad\dfrac{b+a}{b-a}=\dfrac{d+c}{d-c}}[/tex]
Thus,
[tex]\longrightarrow\rm{\dfrac{y-2}{1}=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}}[/tex]
[tex]\Longrightarrow\rm{\dfrac{1+(y-2)}{1-(y-2)}=\dfrac{(e^x+e^{-x})+(e^x-e^{-x})}{(e^x+e^{-x})-(e^x-e^{-x})}}[/tex]
[tex]\longrightarrow\rm{\dfrac{y-1}{3-y}=\dfrac{2e^x}{2e^{-x}}}[/tex]
[tex]\longrightarrow\rm{\dfrac{e^x}{e^{-x}}=\dfrac{y-1}{3-y}}[/tex]
[tex]\longrightarrow\rm{e^{2x}=\dfrac{y-1}{3-y}}[/tex]
[tex]\longrightarrow\rm{(e^x)^2=\dfrac{y-1}{3-y}}[/tex]
Take square root.
[tex]\longrightarrow\rm{e^x=\sqrt{\dfrac{y-1}{3-y}}}[/tex]
Take natural logarithm.
[tex]\longrightarrow\rm{x=\log_e\sqrt{\dfrac{y-1}{3-y}}}[/tex]
[tex]\longrightarrow\rm{x=\dfrac{1}{2}\log_e\left(\dfrac{y-1}{3-y}\right)}[/tex]
Now take [tex]\rm{x=f^{-1}(y).}[/tex]
[tex]\longrightarrow\rm{f^{-1}(y)=\dfrac{1}{2}\log_e\left(\dfrac{y-1}{3-y}\right)}[/tex]
Replace y by x and finally we get,
[tex]\longrightarrow\rm{\underline{\underline{f^{-1}(x)=\dfrac{1}{2}\log_e\left(\dfrac{x-1}{3-x}\right)}}}[/tex]
This is the inverse of the given function.