Answer:
Step-by-step explanation:
If x = 0, 5x = 7x is true.
If x = 1, 5x ≠ 7x
5x = 7x is sometimes true.
which of the following functions is graphed below?
The graphed piecewise function is the one in option B.
Which of the given functions is the graphed one?
On the graph, we can see that on the lowest part we have a closed dot at x = 1
And the above part has a open dot at x = 1.
Then the piecewise function is of the form:
y = f(x) for x ≤ 1y = g(x) for x > 1Such that the above part seems to be quadratic, and the part below seems to have a larger degree.
With that, we can conclude that the correct option is:
y = x^2 + 4, for x > 1y = x^3 - 2, for x ≤ 1Which is option b.
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Solution for the attached question below
The solution to the question is 265.
What is the logarithm of a number?Logarithm of a number A is the exponent or power or index n, a given number called the base B would be raised to give the number A.
So n is the logarithm of A to base B.
Analysis:
log 2( log2(x-9)) = 3
log2(x-9) = [tex]2^{3}[/tex]
log2(x-9) = 8
x-9 = [tex]2^{8}[/tex]
x-9 = 256
x = 256 + 9
x = 265
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The graph of a quartic function crosses the x-axis at -4 and -2 and touches it at 3. State the equation for the family.
The equation of the quartic function is f(x) = x⁴ - 18 · x² + 6 · x + 72.
How to find the possible equations for a quartic polynomial that passes through the x-axis at three points
Herein we must construct at least a polynomial that satisfies all conditions described in the statement. According to the fundamental theorem of algebra, quartic functions may have no real roots, two real roots or four real roots, which means that one of the roots must have a multiplicity of 2.
The root with a multiplicity of 2 is x = 3 and both x = - 4 and x = - 2 have only a multiplicity of 1, then we have the following expression by using the factor form of the definition of polynomials:
f(x) = (x - 3)² · (x + 4) · (x + 2)
Now we expand the expression to get the standard form:
f(x) = (x² - 6 · x + 9) · (x² + 6 · x + 8)
f(x) = x⁴ - 6 · x³ + 9 · x² + 6 · x³ - 36 · x² + 54 · x + 8 · x² - 48 · x + 72
f(x) = x⁴ - 18 · x² + 6 · x + 72
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A university is interested in whether there's a difference between students who live on
campus and students who live off campus with respect to absenteeism. Over one
semester, researchers take random samples of on-campus and off-campus students and
record the following number of missed classes over a semester:
On-campus: (3, 4, 0, 6, 2, 1, 3, 3, 5, 2, 4, 4, 6, 5, 2)
Off-campus: (6, 5, 2, 6, 2, 0, 7, 8, 1, 7, 2, 6, 5, 3, 2)
A. Would we use a t confidence interval or a z confidence interval to determine
whether there's a significant difference between the two groups? What are
the conditions for using this kind of confidence interval? Do these data meet
the necessary conditions? Use sketches of modified box-and-whisker plots to
support your decision. (2 points)
B. What are the degrees of freedom (k) for this test using the conservative
method? (Hint: Don't pool and don't use your calculator.) (1 point)
C. What are the sample statistics for this test? Consider on-campus students to
be sample one and off-campus students to be sample two. (2 points)
D. Compute a 95% confidence interval for the difference between the number of
classes missed by each group of students. (2 points)
E. Compute a 90% confidence interval for the difference between the number of
classes missed by each group of students. (2 points)
F. Based on the two confidence intervals you computed in parts d and e, draw a conclusion about the differences between the means of the two groups.
It is to be noted that the determination of whether or not there is a significant difference between the two groups will be done using a t test.
What is a t test?
A t-test is a statistical test that juxtaposes two samples' means. It is used in hypothesis testing, using a null hypothesis that the variance in group means is zero and an alternative hypothesis that the difference is not zero.
What are the conditions for using this kind of confidence interval?The conditions to use the t test are:
The sample must be independentThe mean of the population and variance must be unknown.The Box plot is attached.What are the degrees of freedom (k) for this test using the conservative method?The degrees of freedom (k) to be utilized for this text will be derived using the conservative method given below:
df = [(s₁²/n₁) + (s₂²/n²)/[((s₁²/n₁)²/((n₁-1)) + (s₂²/n₂)²/((n₂-1))]
= [(3.0952/15) + (6.4095/15)]² / [((3.0952/15)²/14) + ((6.4095/15)²/14)]
= 24.965
Hence,
df ≈ 24 (if approximated to the floor)
What are the sample statistics for this test?Recall the the standard deviation of the population are unequal and unknown. This thus requires that we utilize the two-sample unpooled t-test.
Here, H₀ is given as;
[tex]t = \frac{\bar{x_{1} -\bar{x_{2}}}}{\sqrt{\frac{s_{1}^{2} }{n_{1} } + \frac{S_{2}^{2} }{n_{2}} } } \sim t_{df}[/tex]
t = [(3.33333 - 4.13333)]/√[(3.0952/15) + (6.4095/15)]
= - 0.8/√0.6337
t = - 1.005
What is the 95% confidence interval for the difference between the number of classes missed by each group of students?
The 95% confidence interval is computed using the following formula:
[tex](\bar x_{2} - \bar x_{1}) \pm t_\alpha_/_2_,_df \left({\sqrt{\frac{S_{1}^{2} }{n_{1} } + \frac{S_{2}^{2} }{n_{2}} } } \right)[/tex]
= - 0.8 ± t₀.₀₂₅,₂₄ (√0.6337)
= - 0.8 ± 2.064 (√0.6337)
= -2.4429, 0.8429
What is the a 90% confidence interval for the difference between the number of classes missed by each group of students?To derive the 90% interval, we state:
[tex](\bar x_{2} - \bar x_{1}) \pm t_\alpha_/_2_,_df \left({\sqrt{\frac{S_{1}^{2} }{n_{1} } + \frac{S_{2}^{2} }{n_{2}} } } \right)[/tex]
= - 0.8 ± t₀.₀₅₀,₂₄ (√0.6337)
= - 0.8 ± 0.685 (√0.6337)
= -2.162, 0.562
Based on the two confidence intervals computed in parts d and e, what is the conclusion about the differences between the means of the two groups?
From the intervals computed, we must fail to reject H₀
H₀ : μ₁ = μ₂
It is clear from the above intervals computed from that the differences between the mean of both groups is significant. This is because, zero is included on the two intervals.
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Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options. x < 5 –6x – 5 < 10 – x –6x + 15 < 10 – 5x A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right. A number line from negative 3 to 3 in increments of 1. An open circle is at negative 5 and a bold line starts at negative 5 and is pointing to the left.
The solution to the inequality expression is x>5 and the correct representation is a number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
Inequality expressionInequality expressions are expression not separated by an equal sign
Given the following inequality expression
–3(2x – 5) < 5(2 – x)
Expand
-6x + 15 < 10 - 5x
Collect the like terms
-6x + 5x < 10 -15
-x < -5
x > 5
Hence the solution to the inequality expression is x>5 and the correct representation is a number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
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x minus 2 is equal to 7
Write an equation of the line that passes through the point (4, –5) with slope 2.
answers:
A. y−4=2(x+5)
B. y+5=−2(x−4)
C. y−4=−2(x+5)
D. y+5=2(x−4)
Answer:
D. y+5=2(x−4)
Step-by-step explanation:
The point-slope form of the equation of a line is useful for writing an equation for a line through a given point with a given slope.
Point-slope formThe point-slope form of the equation of a line is ...
y -k = m(x -h) . . . . . . line through point (h, k) with slope m
ApplicationWe want a line through point (h, k) = (4, -5) with slope m = 2. Putting these numbers into the form gives ...
y -(-5) = 2(x -4)
y +5 = 2(x -4) . . . . . . simplifying the signs
Name the figure below in two different ways.
Y
Symbol:
and
M
A
The line segment in the figure can be named as: IYM and MYI.
How to Name a Line Segment?If we have three points on a line segment, the point on the middle will be the alphabet in the center when naming the line, while the alphabets of both endpoints can be written either at the beginning or ending.
Given the figure above, Y will be at the center. The endpoints are I and M. Therefore, the line segment can be named as: IYM and MYI.
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A pitcher originally contains a juice drink with 20 percent cranberry juice. After 4 ounces of cranberry juice is added, the new drink is one-fourth cranberry juice. How many total ounces of juice drink are in the pitcher after the addition of the cranberry juice?
36-u=261 solve for u
Answer:
u = -225
Step-by-step explanation:
36 - u = 261
-u = 261 - 36
-u = 225
u = -225
Last PreCalc Question, Need help with writing piecewise functions with graphs. Giving brainliest!
The piece-wise linear functions can be written as follows:
[tex]f(x) = x, x \leq -2[/tex].[tex]f(x) = -x - 7, -2 < x \leq 1[/tex].[tex]f(x) = 2x - 9, x > 1[/tex].What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.For x equal or less than -2, the line passes through (-3,-3) and (-2,-2), hence the rule is:
[tex]f(x) = x, x \leq -2[/tex].
For x greater than -2 up to 1, the y-intercept is of -7, and the line also passes through (1,-8), hence the rule is:
[tex]f(x) = -x - 7, -2 < x \leq 1[/tex].
For x greater than 1, the function goes through (2,-5) and (3,-3), hence the slope is:
m = (-3 - (-5))(3 - 2) = 2.
The rule is:
y = 2x + b.
When x = 2, y = -5, hence:
-5 = 2(2) + b
b = -9.
Hence:
[tex]f(x) = 2x - 9, x > 1[/tex].
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I need help with my work
The area of the interior above the polar axis is -0.858 square units
The area bounded by a polar curveThe area bounded by a polar curve between θ = θ₁ and θ = θ₂ is given by
[tex]A = \int\limits^{\theta_{2} }_{\theta_{1} } {\frac{1}{2}r^{2} } \, d\theta[/tex]
Now, since we have the curve r = 1 - sinθ and we want to find the area of the interior above the polar axis, we integrate from θ = 0 to θ = π, since this is the region above the polar axis.
So, [tex]A = \int\limits^{\theta_{2} }_{\theta_{1} } {\frac{1}{2}r^{2} } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}(1 - sin\theta)^{2} } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}[1 - 2sin\theta + (sin\theta)^{2}] } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}\, d\theta - \int\limits^{\pi}_{0} 2sin\theta \, d\theta+ \int\limits^{\pi}_{0} (sin\theta)^{2} } \, d\theta\\[/tex]
[tex]A = \int\limits^{\pi}_{0} {\frac{1}{2}\, d\theta - 2\int\limits^{\pi}_{0} sin\theta \, d\theta+ \int\limits^{\pi}_{0} \frac{(1 - cos2\theta)}{2} } \, d\theta\\= \int\limits^{\pi}_{0} {\frac{1}{2}\, d\theta - 2\int\limits^{\pi}_{0} sin\theta \, d\theta+ \int\limits^{\pi}_{0} \frac{1}{2} } \, d\theta -\int\limits^{\pi}_{0} \frac{cos2\theta}{2} } \, d\theta\\[/tex]
[tex]A = \int\limits^{\pi}_{0} \, d\theta - 2\int\limits^{\pi}_{0} sin\theta \, d\theta -\int\limits^{\pi}_{0} \frac{cos2\theta}{2} } \, d\theta\\= [\theta]_{0}^{\pi} - 2[-cos\theta]^{\pi} _{0} - [\frac{sin2\theta}{4}]_{0}^{\pi} \\= [\theta]_{0}^{\pi} + 2[cos\theta]^{\pi} _{0} - [\frac{sin2\theta}{4}]_{0}^{\pi}\\= [\pi - 0] + 2[cos\pi - cos0] - \frac{ [sin2\pi - sin0]}{4}\\= \pi + 2[-1 - 1] - \frac{ [0 - 0]}{4}\\= \pi + 2[-2] - \frac{ [0]}{4}\\= \pi - 4 - 0\\= \pi - 4\\= 3.142 - 4\\= -0.858[/tex]
So, the area of the interior above the polar axis is -0.858 square units
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46.Find the quotient and remainder
Answer:
30624; 1514,6; a1,6.
tiff basic insurance cost is $613.50 per year. His insurance company offers a typical "safe-driver discount." What would be his rate if he went three years without an accident? Annual premium?
The rate if he went three years without an accident based on the information about the insurance is $552.15.
How to calculate the insurance?From the information given, his basic insurance cost is $613.50 per year and his insurance company offers a typical "safe-driver discount.
The amount will be:
= 613.50 - (20.45 × 3)
= 552.15
In conclusion, the rate is $552.15.
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Assume a = b and c ≠ 0
Which of the following sentences is not true?
1) a+c=b+c
2) a-c=b-c
3) ac = bc
4) a/c = b/c
5) none of these
Answer:
None of these is not true
An exponential function f ( x ) = a b x f ( x ) = a b x passes through the points (0, 10000) and (3, 2160). What are the values of a and b ?
The values of a and b of the exponential function are 10000 and 0.6 respectively
How to solve exponential functions?We are given that the exponential function is expressed in general form as; f(x) = abˣ
where;
a is a non-zero real number called the initial value
b is any positive real number such that
b ≠ 1.
The domain of f is all real numbers.
The range of f is all positive real numbers if a > 0.
The range of f is all negative real numbers if a < 0.
The y-intercept is (0, a)
The horizontal asymptote is; y = 0.
We are told that this exponential function passes through the coordinate points (0, 10000) and (3, 2160).
At coordinate point (0, 10000), we have;
10000 = ab⁰
a = 10000
Now, at the coordinate point (3, 2160), we have;
2160 = 10000(b)³
2160/10000 = b³
0.216 = b³
b = ∛0.216
b = 0.6
Thus, we can conclude that the values of a and b of the given exponential function are respectively 10000 and 0.6.
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Two-thirds of a number x is at least 10. Find the smallest possible prime number x.
The smallest possible prime number x is 17.
What is Fraction?An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a correct fraction is smaller than the denominator.
given, Two-thirds of a number x is at least 10.
So,
2/3 x ≥ 10
x ≥ 15
First prime number that is bigger than 15 is 17.
therefore, The smallest possible prime number x is 17.
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Compute the monthly payments for the add on interest loan. The amount of the loan is $8,276.17. The annual interest rate is 5.7%. The term of the loan is 5.5 years.
The monthly payments for this add on interest loan are of $164.71.
Given Information and Formula Used
It is given that for an add on interest loan,
Principal Amount, p = $8,276.17
Annual Interest Rate, r = 5.7%
Term of the loan, T = 5.5 years
The formula for simple interest is given as follows,
I = (p)(r)(t)/100 ............... (1)
The formula for total amount of add on interest is given by,
A = p + I ....................... (2)
Computing the Interest
Substitute the given values of p, r, and t in the formula (1) of interest to get,
I = (8276.17)(5.7)(5.5)/100
I = 259457.9295/100
I = $2,594.58
Computing the Monthly Payment for Add-on Interest Loan
Substituting the values of p and I in the formula (2), we obtain the total amount as,
A = $ (8276.17 + 2594.58)
A = $ 10,870.75
Monthly payment for the add on interest loan = A/t(in months)
= $ (10,870.75/66)
= $164.71
Therefore, monthly payments of $164.71 are to be made for the add on interest loan.
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ABCD is a rectangle, with M the midpoint or BC and N the midpoint of CD. If CM=4 and NC=5, what percent of the area of the rectangle is shaded
Answer:
87.5%
Step-by-step explanation:
Given:
⇒ ABCD is a rectangle, with M the midpoint of BC and N the midpoint of CD.⇒ CM = 4 and NC = 5.Area of triangle NCM
⇒ 1/2 × base × height
⇒ 1/2 × CN × CM
⇒ 1/2 × 5 × 4
⇒ 10cm²
Length of rectangle ⇒ 10
Breadth of the rectangle ⇒ 8
↓
Area of rectangle ⇒ Length × Breadth
Area of rectangle ⇒ 10 × 8 = 80cm²
Area of shaded region ⇒ Area of rectangle - Area of triangle
Area of shaded region ⇒ 80 - 10 = 80cm²
Percentage of the shaded region = Shaded/complete rectangle × 100
⇒ 70/80 × 100 = 87.5
⇒ The percent of the area of the rectangle shaded is therefore 87.5%.
Your last five customer interactions lasted 2, 3, 6, 8, and 4 minutes."
Employee: "That means I've averaged __________ minutes across those 5 interactions."
Answer:
4.6 minutes
Step-by-step explanation:
Average = the sum of the numbers/ the number of numbers
Plug in the numbers: 2+3+6+8+4/5 = 23/5 = 4.6 minutes
What is the current yield on a 3 year bond with 10% annual coupons, a par value of 100, and a current price of 107.87?
The current yield of the bond is 9.27%.
What is current yield?
Current yield is the bond rate of return computed as the annual coupon divided by the current price of the bond
The annual coupon is the coupon rate of 10% multiplied by the bond's par value of 100
annual coupon=10%*100
annual coupon=10
current bond price=107.87
current yield=annual coupon/current bond price
current yield=10/107.87
current yield=9.27%
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f(x)=2x. If g(x) is a vertical stretch, compression, and or reflection of f(x) followed by a, what is the equation of g(x)?
The function g(x) is g(x)= (3x)^2
How to solve for g(x)?
The complete question is in the image
From the graph in the image, we have:
f(x) = x^2
The function f(x) is stretched by a factor of 3 to form g(x).
This means that:
g(x) = f(3x)
So, we have:
g(x)= (3x)^2
Hence, the function g(x) is g(x)= (3x)^2
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If 4x+2=5y+3 then y =
Answer:4x/5-1/5 = y
Step-by-step explanation:
We need to get y by itself
4x+2=5y+3
subtract 3 from both sides
4x-1=5y
divide both sides by 5
4x/5-1/5 = y
Answer:
[tex]4x + 2 = 5y + 3[/tex]
[tex]5y + 3 = 4x + 2[/tex]
[tex]5y = 5x + 2 - 3[/tex]
[tex]5y = 4x - 1[/tex]
[tex] \frac{5y}{5} = \frac{4x - 1}{5} [/tex]
[tex]y = \frac{4x - 1}{5} [/tex]
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 7.9 years, and standard deviation of 0.9 years.
If you randomly purchase one item, what is the probability it will last longer than 9 years?
If you randomly purchase one item, the probability it will last longer than 9 years is; 11.12%
How to find the probability from z-score?
The formula for Z-score is;
z = (x' - µ)/σ
where;
x' = sample mean
µ = population mean
σ = standard deviation
Thus;
z = (9 - 7.9)/0.9
z = 1.22
From online p-value from z-score calculator, we have;
probability = 0.1112 = 11.12%
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The perimeter of a regular hexagon is 72 inches. Find the length of each of its sides.
12 inches
Step-by-step explanation:A regular polygon is any 2-D shape where all the sides and angles are congruent.
Regular Hexagon
As denoted by the prefix "hexa-", hexagons have 6 sides and internal angles. The perimeter is equal to all 6 sides added together. However, we know that these sides must be congruent because it is a regular hexagon.
Solving for Perimeter
We can create an equation that describes the perimeter of this shape, with "x" representing the length of one side.
6x = 72Since all the sides are equal we can use multiplication instead of addition. To solve this equation, divide both sides by 6.
x = 12This means that all sides must be 12 inches.
What are the length and width of a rectangle if the length is
3 inches longer than twice the width and the area of the
rectangle is 5 in2?
The length and width of the rectangle are 5 inches and 1 inches respectively.
How to find the length and width of a rectangle?The length and width of the rectangle can be found as follows;
l = 3 + 2w
area of a rectangle = lw
where
l = lengthw = widthTherefore,
5 = lw
5 = (3 + 2w)w
5 = 3w + 2w²
2w² + 3w - 5 = 0
Hence,
2w² + 3w - 5 = 0
Therefore,
w = 1 and w = - 5 / 2
width = 1 inches
length = 3 + 2(1) = 5 inches
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In 2016, Alberta had about 4.2 million people.
Assuming they follow the same population
growth rate, it is predicted they will have 6.65
million people in 20 years. At what rate is the
province's population growing?
The province's population is growing at the rate of 58.34 %
Rate of change is used to mathematically describe the percentage change in value over a defined period of time, and it represents the momentum of a variable. The calculation for ROC is simple in that it takes the current value of a stock or index and divides it by the value from an earlier period.
Given:
Initial Population = 4.2 million
Population after 20 years = 6.65 million
Change in population = 6.65 - 4.2 = 2.45 million
Rate at which the province's population growing is
= [tex]\frac{Change in population}{Inital Population}[/tex] x100 %
= [tex]\frac{2.45}{4.2}\\[/tex] x 100%
= 58.34 %
Thus the province's population is growing at the rate of 58.34 %
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Use the information provided to write the general conic form equation of the circle: Ends of a diameter: (11, -2) and (9, 4)
The equation for a circle is:
[tex](x-a)^{2} + (y-b)^{2} + = r^{2}[/tex]
Where (a,b) is the circle's center and r is the circle's radius.
First, we can find the center point of the circle. Because the two points from the problem are the endpoints of a diameter, the midpoint of the line segment is the center point of the circle.
The formula to find the mid-point of a line segment giving the two endpoints is:
M = [tex](\frac{x_{1} + x_{2} }{2}[/tex], [tex]\frac{y_{1} + y_{2} }{y} )[/tex]
Where M is the midpoint and the given points are:
[tex](x_{1}, y_{1} )[/tex] and [tex](x_{2} , y_{2})[/tex]
Substituting the values from the two points in the problem gives:
[tex]M = (\frac{11+9}{2}[/tex],[tex]\frac{-2+4}{2} )[/tex]
[tex]M = (\frac{11+9}{2}[/tex],[tex]\frac{2-4}{2})[/tex]
[tex]M = (\frac{20}{2} , \frac{2}{2})[/tex]
[tex]M = (10,1)[/tex]
Find the sum of the geometric series given a1=−1, r=2, and n=7
A. -127
B. -116
C. 1
D. -118
Answer:
A
Step-by-step explanation:
the sum to n terms of a geometric series is
[tex]S_{n}[/tex] = [tex]\frac{a_{1}(r^{n}-1) }{r-1}[/tex] , then for n = 7
S₇ = [tex]\frac{-1(2^{7}-1) }{2-1}[/tex] = [tex]\frac{-1(128-1)}{1}[/tex] = - 1 (127) = - 127
The Rangers won five of the first six games. how many of the next 30 games must the rangers wins have twice as many wins as losses?
Using proportions, it is found that they must win 19 of their next 30 games.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
We want a win-to-loss proportion of 2:1, which is equivalent to a win-to-total proportion of 2:3. We have that:
The team will have won 5 + x games.The team will have played 36 games.Hence:
[tex]\frac{5 + x}{36} = \frac{2}{3}[/tex]
3(5 + x) = 72
15 + 3x = 72
3x = 57
x = 19.
The Rangers must win 19 of their next 30 games.
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