The end behavior of a polynomial function can be determined by examining the leading coefficient and degree of the polynomial. If the leading coefficient is positive, the graph will increase without bound as the x-values increase. If the leading coefficient is negative, the graph will decrease without bound as the x-values increase.
The end behavior of a polynomial function depends on the equation of the function. The degree of a polynomial is the highest power of the variable, and the leading coefficient is the coefficient of the term with the highest degree. If the leading coefficient is positive, the graph will increase without bound as the x-values increase. This is because the terms in the equation with positive coefficients will become increasingly larger than the terms with negative coefficients as the x-values increase. On the other hand, if the leading coefficient is negative, the graph will decrease without bound as the x-values increase. This is because the terms in the equation with negative coefficients will become increasingly larger than the terms with positive coefficients as the x-values increase. In addition, the degree of the polynomial can also affect the end behavior of the graph. If the degree is even, the graph will not have an extreme point. However, if the degree is odd, the graph will have an extreme point. The extreme point will be a maximum or a minimum depending on the sign of the leading coefficient.
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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.
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
someone please help me to solve this problem.
Answer:
PA = PB
Step-by-step explanation:
The radius OA = OB
so triangles OPA is congruent to OPB because OA = OB, OP is the same side, angle OPA = angle OPB = 90o
therefore PA = PB
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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What is the slope of 3?
The slope of the line y = 3 is equal to 0.
The inclination of the supplied line with respect to the x-axis is known as the slope of any line.
The findings indicate that y = 3.
As well known,
the equation for a straight line is y = mx + c.
Where 'c' is the y-axis intercept and 'm' is the slope on the y-axis.
We obtain m = 0 and c = 3 by comparing equation (1) with the standard equation of the straight line.
The formula for a horizontal line is y = 3. Since the y coordinate has not changed, the slope is zero.
The slope of the line y = 3 is therefore equal to 0.
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can any one help in these questions
Answer:
1) x = 10
m<LOH = 55
m<GOH = 85
m< KOG = 40
2) x = 3
m<CQR = 22
m<BQC = 53
m< AQB = 33
m< PQA = 72
3) x = 45
m<TYZ = 41
m<XYT = 49
m< SYX = 90
m< XYZ = 90
4) x = 7
m < KJL = 30
m< NJK = 50
m< MJL = 56
Step-by-step explanation:
1)
Straight line = 180°
So
4x + 15 + 9x - 15 + 4x = 180
17x + 10 = 180
x = (180-10)/17
x = 10
m<LOH = 40
m<GOH = 85
m< KOG = 55
2)
Straight line = 180°
So
7x + 1 + 56 - x + x + 30 + 24x = 180
31x + 87 = 180
x = (180 - 87) / 31
x = 3
m<CQR = 72
m<BQC = 33
m< AQB = 53
m< PQA = 22
3)
Straight line = 180°
So
2x + x + 4 + x - 4 = 180
4x = 180
x = 45
m<TYZ = 41
m<XYT = 49
m< SYX = 90
m< XYZ = 90
4)
Straight line = 180°
So
2(3x + 4) + 11x - 3 + 8x = 180
6x + 8 + 19x -3 = 180
25x + 5 = 180
x = (180 -5) / 25
x = 7
m < KJL = 30
m< NJK = 50
m< MJL = 56
I hope my answer helps you.
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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Define variables for number of 4 1/2 feet pieces of rope and the amount of the 48 feet long rope that’s left over then write an equation to represent the real ashton’s hip between these variables.
9/2 x +48 is an equation to represent the real ashton’s hip between these variables.
To write this type of fraction = 4 + 1/2 - The integer 4 in the expression is the amount of feet of rope that was cut.
Thus to know the number of 1/2 foot pieces produced, let this number be x. Thus, we have 1/2 of x pieces = 2
x/2 = 4; x = 2x4 = 8
Thus, it has 8 pieces of 1/2 foot long piece each of rope.
9/2 x +48 is an equation to represent the real ashton’s hip between these variables.
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There are 36 kids in the class.
1/4
of the kids are teenagers.
1/3
of the teenagers are wearing leggings.
How many kids in the class are teenagers wearing leggings?
The number of kids in the class teenagers wearing leggings is 3.
What is a fraction?In such a fraction, the value above the horizontal line is referred to as the numerator.
In another word, if we divide two numbers by each other then the upper word is called the numerator, and the lower word is called the denominator.
As per the given,
Number of kids = 36
Fraction of kids are teenagers = 1/4 of 36
(1/4)36 = 9 kids
Fraction of teenagers wear leggings = 1/3 of 9
(1/3)9 = 3 kids
Hence "There are three students in the class who are teenagers and wearing leggings".
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if p and q both are prime numbers what cannot be the difference between them?
A: 1
B: 23
C: 3
D: 9
Answer:
B
Step-by-step explanation:
prime number ( 1 ≤ x ≤ 30 )
= {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
for answer choice A: 3-2 = 1
C: 5-2 = 3
D: 11-2 = 9
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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To study the effect of temperature on yield in a chemical process, five batches were produced at each of three temperature levels. The results follow. Construct an analysis of variance table. Use a. 05 level of significance to test whether the temperature level has an effect on the mean yield of the process.
Temperature
50° C60° C70°C
34 30 23
24 31 28
36 34 28
39 23 30
32 27 31
Construct an analysis of variance table is given below in the answer.
What is tempreture?Temperature in mathematics is a measure of how hot or cold something is compared to a reference point. Temperature is a scalar quantity, meaning it has magnitude but no direction. Temperature is measured in units such as Celsius, Fahrenheit, and Kelvin. Temperature is typically associated with thermodynamic systems in physics, which can be modeled mathematically.
Analysis of Variance Table
Source of Variation Sum of Squares Degrees of Freedom Mean Square F-test Significance Level
Temperature 65.6 2 32.8 5.14 0.05
Error 57.4 12 4.78
Total 123 14
The F-test statistic (5.14) is greater than the critical value of 3.89, indicating that the temperature level has a significant effect on the mean yield of the process at the 0.05 level of significance.
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Rajan bought a book for Rs 180 and sold it to Sajan
at a profit of 20%. Sajan sold that book to Nirajan
at a loss of 20%. At what price Nirajan should sell
the book to receive 5% profit?
Ans: Rs 181.44
The book should be sold at Rs.181.49 to receive a 5% profit.
What are percentages?In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is one of the ways to represent a dimensionless relationship between two numbers; other methods include ratios, fractions, and decimals. Percentages are often denoted by the symbol "%" written after the number.
Given here, CP for Sajan = 180
and SP for Sajan = (1+20%)
= 1.2 × 180
= 216
CP for Niranjan = 216 and SP = 0.8 × 216
= 172.8
CP for Niranjan = 172.8
therefore to sell it at 5% profit we get SP for Niranjan = 172.8 × 1.05
= 181.49
Hence selling price for Niranjan is equal to Rs.181.49
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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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write the equation of a line in slope-intercept form that is parallel to the line y=7x-8
Answer:
Below
Step-by-step explanation:
Replace the '8' with any number
here is one
y = 7 x + 4
keegan bought 5 T-shirts and 2 pairs of shoets for $49.50. Each pair of shorts, s, cost twice as much as each T-shirt, t.
Part A
Which system ot equations can vevused to find the price of the shorts and t-shirts?
(use attachment)
Part B
how much did keegan pay for each t-shirt and each pair of shorts?
Answer:
A) [tex]{2s + 5t = 49.50\\\\s = 2t[/tex]
B) Keegan paid $5.50 for each t-shirt and $11 for each short.
Step-by-step explanation:
A)
⭐ What is a system of equations?
A system of equations is two or more equations that intersect at a common point (x,y)We are asked to create a system of equations for Keegan's situation.
Let s = the price for every short
Let t = the price for every t-shirt
So, if Keegan buys 2 shorts and 5 t-shirts, his cost will be $49.50.
As an equation, this situation would be:
[tex]2s + 5t = 49.50[/tex]
It's given that the cost of 1 short is twice the cost of 1 t-shirt.
As an equation, this situation would be:
[tex]s = 2t[/tex]
B)
⭐How can we solve a system of equations?
You can solve a system of equations by substitutionYou can solve a system of equations by graphingYou can solve a system of equations by eliminationFor this problem, the best way to solve this system of equations is through substitution. I will demonstrate how to use substitution for this problem.
Using the equations [tex]2s + 5t = 49.50[/tex] and [tex]s = 2t[/tex], we will literally substitute a variable from one equation into another equation to solve for the other variable.
Given that [tex]s = 2t[/tex]:
[tex]2s + 5t = 49.50[/tex]
[tex]2(2t) + 5t = 49.50[/tex]
[tex]4t + 5t = 49.50[/tex]
[tex]9t = 49.50[/tex]
[tex]t = 5.50[/tex]
∴ It costs Keegan $5.50 for one t-shirt.
Now, we can substitute the value of t into the same equation to solve for the value of s.
Given that [tex]t = 5.50[/tex]:
[tex]2s + 5(5.50) = 49.50[/tex]
[tex]2s + 27.5 = 49.50[/tex]
[tex]2s = 22[/tex]
[tex]s = 11[/tex]
∴ It costs Keegan $11 for one short.
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
What is the base of log(x) = 100? Explain.
Answer:
10
Step-by-step explanation:
If no base is written, the common logarithm is implied.
Answer:
the value of x will depend on the base of the logarithm.
Step-by-step explanation:
The base of a logarithm is the number that is used as a factor in the exponential equation that defines the logarithm. For example, in the logarithmic equation log10(x) = 100, the base is 10.
In the equation log(x) = 100, the base of the logarithm is not given. In order to solve for x, we need to know the base of the logarithm. Without this information, we cannot find the value of x.
For example, if the base of the logarithm is 10, we can rewrite the equation as log10(x) = 100, and then use the following property of logarithms to solve for x:
log10(x) = 100
10^(log10(x)) = 10^100
x = 10^100
On the other hand, if the base of the logarithm is 2, we can rewrite the equation as log2(x) = 100, and then use the following property of logarithms to solve for x:
log2(x) = 100
2^(log2(x)) = 2^100
x = 2^100
a shop ordered 13 boxes marbles each box contains six 10s of marbles each tin holds 45 marbles how many marbles does a shop order
Answer:
3510 marbles
Step-by-step explanation:
45 x 6 = 270
270 x 13 = 3510
Hope this helps :)
Answer:3510
Step-by-step explanation:
:)
What is the nature of roots of the equation 4x² 4x 1 0?
The nature of roots of the equation 4x² -4x -19 = 0 is : The roots are real.
Given quadratic equation to us is 4x² -4x -19 = 0.
With respect to Standard form ax² +bx +c = 0 , Discriminant of the quadratic equation is given by b² - 4ac . Here's a table for Nature of Roots .
SI. NO. CONDITION NATURE OF ROOTS
1 Discriminant > 1 Real roots
2 Discriminant = 1 Equal roots
3 Discriminant < 1 Complex roots
Hence here , with respect to Standard form ,
• a = 4 , • b = -4 & •c = -19 .
⇒ Discriminant = b² - 4ac
⇒ Discriminant = (-4)² - 4 (4) (-19)
⇒ Discriminant = 16 + 304
⇒ Discriminant = 320
Hence since Discriminant is Greater than 0 , hence the nature of roots of the quadratic equation is real.
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1/6 x = 12
HELP PLEASE
Answer:
x=72
Step-by-step explanation:
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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According to the rational root theorem what are all the potential rational roots of f(x)=15x^11-6x^8.
The potential rational roots must be factors of -6/15. These factors are ±1, ±2, ±3, ±6, and ±10.
The rational root theorem states that all rational roots of a polynomial equation must be factors of the constant term divided by factors of the leading coefficient.
In the case of the polynomial f(x)=15x^11-6x^8, the constant term is -6 and the leading coefficient is 15. This means that all potential rational roots must be factors of -6/15. These factors are ±1, ±2, ±3, ±6, and ±10. Therefore, the potential rational roots of f(x)=15x^11-6x^8 are ±1, ±2, ±3, ±6, and ±10.
The rational root theorem is an important tool for solving polynomial equations. It allows us to quickly determine all the potential rational roots of a polynomial equation, which can then be used to solve the equation. In the case of f(x)=15x^11-6x^8, the potential rational roots can be used to determine the factorization of the polynomial. For example, if we were to factorize f(x) using the potential rational roots ±1, ±2, ±3, ±6, and ±10, we would get[tex](x+1)(x-1)(x+2)(x-2)(x+3)(x-3)(x+6)(x-6)(x+10)(x-10)[/tex]. This factorization can then be used to solve the equation.
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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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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:
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.
What is the end behavior of the graph of the polynomial function y 10x9 4x?
The end behavior of the graph of the polynomial function y = 10x^9 + 4x is asymptotic to negative infinity as x approaches negative infinity, and asymptotic to positive infinity as x approaches positive infinity.
The end behavior of a polynomial function is determined by the degree of the polynomial. The polynomial function y = 10x^9 + 4x is a degree 9 polynomial, so it will exhibit an asymptotic behavior in which the graph approaches either positive or negative infinity as x approaches either positive or negative infinity. In this case, the graph will approach negative infinity as x approaches negative infinity, and it will approach positive infinity as x approaches positive infinity. This behavior is due to the fact that the highest degree term (10x^9) will dominate the behavior of the graph as x gets very large in either the positive or negative direction. The lower degree terms (4x) will have a smaller effect on the end behavior of the graph.
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What are the factors of 3x² 4x 4?
The factors of 3[tex]x^{2}[/tex] -4x -4 is (3x+2)(x-2)
Let's start by putting empty brackets.
()
We know that 3[tex]x^{2}[/tex] is 3x*x
Each of these can be inserted into a particular bracket.
(3x)(x)
The factors of -4 are then examined. How many different methods are there to multiply two numbers to reach -4?
1 × -4 = -4
2 × -2 = -4
Choosing which pair to utilize and where to place each is now the challenging part. You might need to try each one separately at first, but with practice this will become simpler. I'll begin with the right couple, which are +2 and -2.
(3x + 2)(x - 2)
We can enlarge it once again to confirm whether this is accurate. Do not forget to multiply each value in the second bracket by each value in the first bracket
so, 3x *x = 3[tex]x^{2}[/tex]
3x * -2 = -6x
Then
2 * x =2x
2 * -2 = -4
All of these together gives us 3[tex]x^{2}[/tex] - 6x +2x -4 which is 3[tex]x^{2}[/tex] - 4x -4
If we had reversed the factors by putting (3x−2)(x+2), it would have given us 3[tex]x^{2}[/tex] - 4x -4, which is close but not exactly what we are looking for.
If the alternative set of components, 1 and -4, had been employed, we would have had ( 3 x + 1 ) ( x- 4 ), which would be 3[tex]x^{2}[/tex]- 11x -4
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Is a discontinuity the same as a hole?
A discontinuity is a point at which a function is not continuous. This means that there is a break or gap in the function at that point, which can be caused by a variety of factors such as a sudden change in the value of the function, a division by zero, or an undefined value.
A function can have a "hole" if it is not defined at a certain input value. For example, consider the following function:
f(x) = 1/x
This function has a hole at x=0 because it is not defined at that value. If you try to evaluate the function at x=0, you will get an error or an undefined result.
A hole is a missing or incomplete part of a function. For example, a function may have a hole at a certain point if the function is not defined at that point, or if there is a gap in the graph of the function at that point.
So, while a discontinuity is a break or gap in a function, a hole is a missing or incomplete part of the function. These two concepts are related, but they are not the same thing.
To determine whether a function has a hole, you can consider the domain of the function, which is the set of all input values for which the function is defined. If there are any values in the domain for which the function is not defined, then the function has a hole at those values.
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f(x) = 1/x
Because it is not defined at x=0, this function has a gap there. You will encounter an error or an undefined outcome if you attempt to evaluate the function at x=0.
A hole is a component of a function that is absent or inadequate. For instance, if a function is not defined at a particular point or if there is a gap in the function's graph at that location, the function may have a hole there.
Therefore, a hole is a missing or unfinished portion of the function, whereas a discontinuity is a break or gap in the function. Although these two ideas are distinct from one another, they are linked.
If 64^x=1/256^y then the value of 3x+4y is
Given,
[tex]\longrightarrow\rm{64^x=\dfrac{1}{256^y}}[/tex]
We have to find out the value of 3x + 4y.
Multiplying both sides by [tex]\rm{256^y}[/tex] we get,
[tex]\longrightarrow\rm{64^x\cdot256^y=1}[/tex]
Then we will write 64 and 256 as powers of 4 as,
[tex]\rm{64=4^3}[/tex]
[tex]256=4^4[/tex]
Then,
[tex]\longrightarrow\rm{(4^3)^x\cdot(4^4)^y=1}[/tex]
[tex]\longrightarrow\rm{4^{3x}\cdot4^{4y}=1\quad\quad[\because(a^m)^n=a^{mn}]}[/tex]
[tex]\longrightarrow\rm{4^{3x+4y}=1\quad\quad[\because a^m\cdot a^n=a^{m+n}]}[/tex]
Now we will take log to the base 4.
[tex]\longrightarrow\rm{\log_4(4^{3x+4y})=\log_4(1)}[/tex]
[tex]\small\text{$\longrightarrow\rm{(3x+4y)\log_4(4)=\log_4(1)\quad\quad[\because\log_b(a^m)=m\cdot\log_b(a)]}$}[/tex]
We have,
[tex]\rm{\log_4(4)=1}[/tex]
[tex]\log_4(1)=0[/tex]
Then,
[tex]\longrightarrow\rm{(3x+4y)(1)=0}[/tex]
[tex]\longrightarrow\rm{\underline{\underline{3x+4y=0}}}[/tex]
Hence 0 is the answer.