1. Find the domain and range of the function
f(x)=√1-4x².
Answers
[tex]f(x) = \sqrt{1 - 4{ x}^{2} } [/tex]
[tex]1 - 4x {}^{2} \geqslant 0[/tex]
[tex] - \infty \: \: \: \: \: \: \: \: - 0.5 \: \: \: \: \: \: \: \: \: 0.5 \: \: \: \: \: \: \: \: \infty \\ - \: \: \: \: \: \: \: \: \: \: \: \: \: \:0 \: \: \: \: \: + \: \: \: \:0 \: \: \: \: \: - [/tex]
[tex]domain \\ [ \: -0.5 \: , \: 0.5 \: ][/tex]
[tex]g(x) = 1 - 4x {}^{2} \\ g'(x) = - 8x \\ g'(x) = 0 \\ x = 0 \\ g(0) = 1 - 4(0) = 1 \\ maximum \: = 1[/tex]
[tex]range \\ [ \: 0 \: , \: 1 \: ][/tex]
Answer:
[tex]\textsf{Domain}: \quad \left[-\dfrac{1}{2}, \dfrac{1}{2}\right][/tex]
[tex]\textsf{Range}: \quad [0, 1][/tex]
Step-by-step explanation:
Domain: set of all possible input values (x-values)
Range: set of all possible output values (y-values)
Given function:
[tex]f(x)=\sqrt{1-4x^2}[/tex]
As negative numbers don't have real square roots:
[tex]\implies 1-4x^2\geq 0[/tex]
Therefore, to find the domain, solve the inequality.
Subtract 1 from both sides:
[tex]\implies -4x^2\geq -1[/tex]
Divide both sides by -1 (reverse the inequality):
[tex]\implies 4x^2 \leq 1[/tex]
Divide both sides by 4:
[tex]\implies x^2\leq \dfrac{1}{4}[/tex]
[tex]\textsf{For }\:a^n \leq b,\:\:\textsf{if }n\textsf{ is even then }-\sqrt[n]{b} \leq a \leq \sqrt[n]{b}:[/tex]
[tex]\implies -\sqrt[2]{\dfrac{1}{4}} \leq x \leq \sqrt[2]{\dfrac{1}{4}}[/tex]
[tex]\implies -\sqrt{\dfrac{1}{4}} \leq x \leq \sqrt{\dfrac{1}{4}}[/tex]
[tex]\implies -\dfrac{1}{2} \leq x \leq \dfrac{1}{2}[/tex]
Therefore:
[tex]\textsf{Domain}: \quad \left[-\dfrac{1}{2}, \dfrac{1}{2}\right][/tex]
To find the range, input the endpoints of the domain into the function:
[tex]\implies f\left(-\dfrac{1}{2}\right)=\sqrt{1-4\left(-\dfrac{1}{2}\right)^2}=0[/tex]
[tex]\implies f\left(\dfrac{1}{2}\right)=\sqrt{1-4\left(\dfrac{1}{2}\right)^2}=0[/tex]
To find the limit of the range, find the extreme point(s) of the function by differentiating the function and setting it to zero.
[tex]\implies f(x)=(1-4x^2)^{\frac{1}{2}}[/tex]
[tex]\implies f'(x)=\dfrac{1}{2}(1-4x^2)^{-\frac{1}{2}} \cdot -8x[/tex]
[tex]\implies f'(x)=-\dfrac{4x}{\sqrt{1-4x^2}}[/tex]
Setting it to zero and solving for x:
[tex]\implies -\dfrac{4x}{\sqrt{1-4x^2}}=0[/tex]
[tex]\implies -4x=0[/tex]
[tex]\implies x=0[/tex]
Substitute x = 0 into the function:
[tex]\implies f(0)=\sqrt{1-4(0)^2}=1[/tex]
Therefore, the range is [0, 1]
Related Questions
Help Me Please with this Math Money Question It Wont Me Screenshot But
Question: CD's are 5 for $30.00.
What is the price of 3 CD's?
Answers:
A) $18.00
B) $15.00
C) $6.00
D) $3.00
Answers
Answer:
A
Step-by-step explanation:
$30.00 ÷ 5 = $6.00
$6.00 × 3 = $18.00
First we need to find out the cost of one CD
We can do that by 30/5 =6
One CD cost 6 $
But the questions ask the cost of 3 CD
So the price for 3 CD is 18
Given angle ABC shown on the coordinate plane below.
Draw angle A”B””C” = Ro 90(T<-4,3>(ABC))
Answers
The attached image shows the image of A"B"C" after the transformation
What is the transformation of the triangle about?The transformation rule states:
A"B"C" = Ro90° (T(-4,3)(ABC))
This implies that one need to rotate the triangle in a 90⁰ clockwise direction, and then one need to translate the triangle.
Using the image shown, the coordinates of ABC are;
A = (-1, 2)
B = (1, 4)
C = (3, -1)
The 90⁰ rule clockwise rotation will be:
(x,y) -- (y,-x)
So, when translated, it will be:
A' = (2, 1)
B' = (4, -1)
C' = (-1, -3)
Then the translation of the triangle using T(-4,3):
(x, y) - (x - 4, y + 3)
So, there is:
A'' = (-2, 4)
B'' = (0, 2)
C'' = (-5, 0)
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The average heart rate of an adult human is 72 beats per minute. For an adult elephant, the average heart rate is 35 beats per minute.
A. Whose heart beats more times in one hour
B. Whose heart makes 1,000,000 beats in less time
Answers
Asap I need a answer please
Answers
Answer:
-4
Step-by-step explanation:
in order to get from -6 to 24, you need to multiply x by -4 since -6* -4 is 24
Company A manufactures and sells gidgets. The owners have determined that the company has the monthly revenue and cost functions shown, such that x represents the number of gidgets sold.
R(x) = 16x
C(x) = 12x + 1,424
At what number of gidgets sold will the company break-even (the point where revenue equals cost)?
Answers
Considering the given equations for revenue and cost, the company will break-even when 356 widgets are sold.
What is the break-even point?The break-even point is the value of x for which:
R(x) = C(x).
In this problem, the functions are:
R(x) = 16x.C(x) = 12x + 1424.Hence:
R(x) = C(x)
16x = 12x + 1424
4x = 1424
x = 1424/4
x = 356
The company will break-even when 356 widgets are sold.
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In the following activity, write an equation to represent each verbal statement, and use it to find the value of each unknown number. Then, put the solution values in order from smallest to largest. [Note: The smallest solution is "first", and the largest solution is "fifth".]
Answers
The solution values in order from smallest to largest are;
-13 - First-10 - Second-3.5 - Third-2 - Fourth-0.75 - FifthUnknown equationlet
the unknown number = x(x - 5)-2 = 14
-2x + 10 = 14
-2x = 14 - 10
-2x = 4
x = 4/-2
x = -2
4x - 5 = -8
4x = -8 + 5
4x = -3
x = -3/4
x = -0.75
(x + 3) / 5 = -2
x + 3 = -2(5)
x + 3 = -10
x = -10 - 3
x = -13
1/2x + 4 = -1
1/2x = -1 - 4
1/2x = -5
x = -5 ÷ 1/2
= -5×2/1
x = -10
x + 2 = 6/-4
-4(x + 2) = 6
-4x - 8 = 6
-4x = 6 + 8
-4x = 14
x = 14/-4
x = -3.5
Therefore, the smallest value is -13 and the largest value is -0.75.
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What is the solution to the following system of equations?
x-3y=6
2x + 2y = 4
(-1,3)
(3,-1)
(1, -3)
(-3,1)
Answers
Answer:
(b) (3,-1)
Step-by-step explanation:
One can determine the correct answer by checking to see if it satisfies the equations.
CheckingThe first equation can be rewritten to ...
x = 6 +3y . . . . . . . add 3y to both sides
This makes it easier to check answer choices:
a) -1 = 6 +3(3) . . . . no
b) 3 = 6 +3(-1) . . . . yes . . . . . (3, -1) is the solution
c) 1 = 6 +3(-3) . . . . no
d) -3 = 6 +3(1) . . . . no
__
Additional comment
We can further convince ourselves this is the correct choice by seeing if it satisfies the second equation:
2x +2y = 4 . . . for (x, y) = (3, -1)
2(3) +2(-1) = 4 . . . . yes
How many triangles are created in the following situation:
m/A=35°, a = 12,b=16
1 triangle
2 triangles
3 triangles
no triangle
Answers
Answer:
Step-by-step explanation:
3 things of triangle are given.
2 sides + 1 angle
so, possible triangles = 1 ans
Assume that the random variable X is normally distributed, with mean μ = 90 and standard deviation σ = 12. Compute the probability P(X < 105).
Answers
The probability P(X < 105), assuming that the random variable X is normally distributed, with mean μ = 90 and standard deviation σ = 12 is calculated to be 0.8944.
When a random variable X is normally distributed, with the mean as μ, and standard deviation as σ, then the probability P(X < x) is calculated using the Z-score table, as the probability P(Z < z), where z is calculated using the formula, z = (x - μ)/σ.
In the question, we are asked to find the probability P(X < 105), assuming that the random variable X is normally distributed, with mean μ = 90 and standard deviation σ = 12.
We calculate as follows:
P(X < 105)
= P(Z < (105-90)/12)
= P(Z < 1.25)
Now we check the z-score table for the value of z as 1.25.
On the left column we choose the value 1.2.
On the top row, we choose the value .05.
The intersection of these, gives us the value,
P(Z < 1.25) = 0.8944.
Thus, the probability P(X < 105), assuming that the random variable X is normally distributed, with mean μ = 90 and standard deviation σ = 12 is calculated to be 0.8944.
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Find the domain of the function. Express the exact answer using interval notation.
f(x) = 2/3x+8
Answers
Answer:
(-oo, -8/3) U (-8/3, oo)
Step-by-step explanation:
Hope this helps.
The domain of the equation f(x) = 2/(3x + 1) is (-∝, -8/3) U (-8/3, ∝)
How to calculate the domain of the equation?From the question, we have the following parameters that can be used in our computation:
f(x) = 2/(3x + 1)
The above equation is a rational function
The rule of a function is that
The domain is the set of all input numbers
For the domain, we set the denominator not equal to 0
So, we have
3x + 8 ≠ 0
Evaluate
x ≠ -8/3
When represented as interval notation, we have
Domain = (-∝, -8/3) U (-8/3, ∝)
Hence, the domain is (-∝, -8/3) U (-8/3, ∝)
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30 feet below the surface of the water. You are ascending towards the
surface of the water. The graph models this situation. What is the slope of the line?
Answers
Answer is slope of 5
Step by step
You can graph the slope as seen on the attached picture. You go up 30, you go right +6. Slope is y/x, so 30/6, equals 5
You can also use the slope formula
(y2-y1) over (x2-x1)
Using points on the graph
(6,0) and (0,-30)
(-30-0 ) over (0-6)
-30 over -6
-30/-6
= 5
The length of the major axis of the ellipse below is 14, and the length of the
red line segment is 3. How long is the blue line segment?
Answers
Answer:
11
Step-by-step explanation:
The focal length is 14, and thus the sum of the lengths of the blue and red segments is also 14.
a two digit number has twice as many ones as tens twice the original number is 9 more than the reversed number find the original number
Answers
Answer: 36
Step-by-step explanation:
As this is a two-digit number, it will have a digit in its 10's place and a digit in its 1's place. We can represent the 10's digit using x and the 1's digit using y.
Since the ones is twice the number of tens, this means that y is equal to 2*x. Let's put this into an equation.
[tex]y=2x[/tex]
If the 10's digit is x and the 1's digit is y, the original number would be [tex]10x + y[/tex], as x must be multiplied by 10 to get it to the 10's place. Similarly, the reversed number would be [tex]10y + x[/tex], as y must be multiplied by 10 to get it to the 10's place.
Twice the original number (10x+y) is equal to 9 more than (i.e. 9 plus) the reversed number. We can put this into an equation to help us answer the question.
[tex]2(10x+y)=9+10y+x\\20x+2y=9+10y+x\\19x+2y=9+10y\\19x=9+8y[/tex]
Now we have a system of two equations.
[tex]y=2x\\19x=9+8y[/tex]
We can solve this system by substituting y for 2x in the second equation. Then, we can isolate and get the value of x.
[tex]19x=9+8(2x)\\19x=9+16x\\3x=9\\x=3[/tex]
Now that we have the value of x, let's put it back into the first equation and solve for y.
[tex]y=2(3)\\y=6[/tex]
Remember that x is the 10's digit and y is the one's digit of our answer. Since x is 3 and y is 6, our answer is 36.
CheckingWe can quickly check if our answer is right by making sure both conditions in the question are met.
6 (the one's digit) is twice the value of 3 (the ten's digit), making the first condition true. Twice of 36 is 72, which is 9 more than the reversed number, which is 63.
The day you were born your Grandparents started to save money for your future education. They decided to deposit $3,000 at the end of each year, for eighteen years, in a savings account that pays 7.5% per year, compounded annually.
On your 19th birthday you get admission into university for a four-year degree course. You calculate that you require approximately $35,000, for your fees, books and living expenses for each year. Assuming you keep the money in the bank accruing the same interest, and withdraw $ 35,000, at the beginning of each year.
a) Calculate the total value of this investment at the end of the term?
b) Determine the total interest earned?
Will your inheritance last you for the full tenure of your four years at the university?
Answers
The total value of the investment is $107,032.16.
The total interest earned is $53,032.16
The inheritance would last the tenure of the university.
What is the total value of the investment?The formula that can be used to determine the future value of the 18-year annuity is: annuity factor x amount deposited yearly
Annuity factor = {[(1+r)^n] - 1} / r
Where:
r = interest rate n = number of years$3000 x [(1.075^18 - 1) / 0.075] = $107,032.16
Amount deposited in the course of 18 years = (3000 x 18) = 54,000
Interest earned = 107,032.16 - 54,000 = $53,032.16
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Find a12 given the geometric sequence 4, 8, 16, 32, ...
A. 32,768
OB. 8192
OC. 16,384
D. 4096
Reset Selection
Answers
The twelfth element of the geometric sequence is equal to 4,096. (Correct choice: D)
How to find a determined element of a geometric sequence by exponential formulae
Sequences are series of elements generated according to at least one condition, usually equations. geometric sequences are generated according to a exponential formulas, whose form and characteristics are described below:
f(n) = a · bⁿ ⁻ ¹ (1)
Where:
a - First element of geometric sequenceb - Common ratio of the geometric sequencen - Element index within the geometric sequenceIf we know that a = 4, b = 2 and n = 12, then the twelfth element of the geometric sequence from the statement is:
f(12) = 4 · 2¹² ⁻ ¹
f(12) = 4 · 2¹¹
f(12) = 4 · 2,048
f(12) = 4,096
The twelfth element of the geometric sequence is equal to 4,096. (Correct choice: D)
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You make a scale drawing of a tree using the scale 5 inches. = 27 feet. If the tree is 67.5 feet tall, how tall is the scale drawing?
Answers
Answer:
12.5 inches
Step-by-step explanation:
We use proportion to calculate the scale drawing
27 feet = 5 inches, so
67.5 feet = more inches (we know that it would be bigger than 5 inches)
We then put the bigger number on top and crossmultiply by saying:
67.5 over 27 × 5
Your answer should come out as 12.5 inches.
Find the measure of x
Answers
Answer:
113°
Step-by-step explanation:
Angles that form a linear pair are supplementary, and thus add to 180°.
10. What is the product of 6% and 14%
Answers
write an equation in slope -intercept form for the line that passes through the points (8,-3) and has a slope of 3 over 2
Answers
Answer:
y = 3/2x -15
Step-by-step explanation:
y = mx + b this is the slope intercept form. We need the slope and the y intercept. The slope (m) has been given (3/2), we just need the b (y-intercept). We will use the point given to find b. The point given is (8,-3). 8 is the x term and -3 is the y term. We will plug them into y = 3/2x + b to solve for b.
y = mx + b
y = (3/2) x + b The slope was given
-3 = (3/2)(8) + b Plugged in the given x and y from the point (8, -3)
-3 = 12 + b I can make 8 into a fraction by putting a 1 under it.
(3/2)(8/1). I can cross cancel to make the number easier to
2 and 8 both share the factor 2 so I will divide both numbers
by 2 and I will now have (3/1)(4/1). 3x4 is 12 and 1x1 is 1, so
12/1 is the same as 12.
-15 = b Subtract both sides by 12. -3-12 is -15
Now that I have the m and the b, I can write the equation in the slope intercept form. y = 3/2 + (-15) or y = 3/2 -15
a infant grew 2/4 inches in the first month 7/4 inches in the third month . first find the total inches the infant grew over the three months . then find the difference in the infants from the second month to the third month
Answers
The fraction computed shows that the total inches the infant grew over the three months is 3 3/4 inches.
How to compute the fraction?First month = 2/4 inches
Second month = 1 inch
Third month = 7/4 inches.
The total inches the infant grew over the three months will be;
= 2/4 + 1 + 7/4
= 3 1/4 inches.
The difference in the infants from the second month to the third month will be:
= 7/4 - 1
= 3/4 inches.
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10 - 13x + 2 in words
Answers
The simplification of the equation above in words is = Twelve minus thirteen X.
Simplification of an equationIn mathematics, coefficients are those numbers that multiplies a variable. For example in 13x, the coefficient is 13 which multiplies a variable known as X.
The equation given is,
10 - 13x + 2
Arrange the digits without a coefficient together.
That is ,
10+2 -13x
12 - 13x
Therefore the simplification of the given equation in words is Twelve minus thirteen X.
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Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 17 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with = 0.40 gram.
(a)
Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)
lower limit
upper limit
margin of error
(b)
What conditions are necessary for your calculations? (Select all that apply.)
normal distribution of weights
is unknown
uniform distribution of weights
is known
n is large
Correct: Your answer is correct.
(c)
Interpret your results in the context of this problem.
The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.
We are 20% confident that the true average weight of Allen's hummingbirds falls within this interval.
We are 80% confident that the true average weight of Allen's hummingbirds falls within this interval.
The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.
Correct: Your answer is correct.
(d)
Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.13 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)
Incorrect: Your answer is incorrect.
hummingbirds
Answers
a)
E =0.124Lower limit =3.026Upper limit = 3.274b) Conditions:[tex]\sigma[/tex] is understood to have a regular distribution of weights
c) Interpretation: 80 percent of the time, this range will include the actual average weight of Allen's hummingbirds, hence its reliability may be counted on
d) sample size [tex]n \approx 27[/tex].
What conditions are necessary for your calculations?Generally,
Here n = 17
T= 3.15
[tex]\sigma=0.40[/tex]
confidence level =c = 0.80
Here we will usethe z critical value.
z critical value for (1+c) /2 = (1+0.80)/2
z critical value for (1+c) /2= 0.9
Zc = 1.28
Critical value = 1.28
a) Margin of error (E) :
[tex]E = Zc*\frac{\sigma}{\sqrt{n}}\\\\\E = 1.28*\frac{0.40}{\sqrt{17}}[/tex]
E =0.124
Margin of error = E =0.124
Lower limit = x-E
L= 3.15 - 0.124
L= 3.026
Lower limit =3.026
Upper limit = x + E
U= 3.15 + 0.124
U= 3.274
Upper limit = 3.274
The following is a confidence interval with a value of 80 percent for the mean weights of Allen's hummingbirds in the area under study: (3.02,3.28)
b) Conditions:[tex]\sigma[/tex] is understood to have a regular distribution of weights
c) Interpretation: 80 percent of the time, this range will include the actual average weight of Allen's hummingbirds, hence its reliability may be counted on.
d) When is already known, the following equation may be used to determine the appropriate number of samples, n:
[tex]n=(\frac{z_c*\sigma }{E})^2[/tex]
Margin of error = 0.13
Sample size (n):
[tex]n= (\frac{z_c*\sigma }{E})^2[/tex]
[tex]n=(\frac{ 1.28*0.36}{0.09})^2[/tex]
n = 26.21
[tex]n \approx 27[/tex]
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Calculate the area of a rectangular field with perimeter 88 cm and length 18 cm
Answers
The area of the rectangular field is 468cm².
Step-by-step explanation:
If, length is 18cm; width is unknown( let x represent width). Then,
Perimeter of a rectangle= Length + Width+ Length + Width
88= 18 + x + 18 + x
88= 36 + 2x
88-36 =2x
52= 2x
52/2=2x/2
26=x
•x=26cm
The width of the rectangle is 26cm.
Area of a rectangle= Length × Width
=18cm×26cm
= 468cm².
The vertices of a quadrilateral are A(-3,-1), B(1,5), C(5,5), and D(5,-1). Select the statement that represents this quadrilateral. A. ABCD is a rectangle because it has exactly one pair of right angles. B. ABCD is a trapezoid because it has at least one pair of parallel sides. C. ABCD is a square because it has all equal sides. D. ABCD is a parallelogram because it has two pairs of parallel sides.
Answers
Answer:
B. ABCD is a trapezoid because it has at least
one pair of parallel sides.
Step-by-step explanation:
the two points A(-3,-1) and D(5,-1) have the same y-coordinates
Then
The line AD is parallel to the x-axis
On the other hand,
the two points B(1,5) and C(5,5) have the same y-coordinates
Then
The line BC is parallel to the x-axis.
We obtain :
• AD is parallel to the x-axis
• BC is parallel to the x-axis.
Therefore
AD // BC
Conclusion:
ABCD is a trapezoid because it has at least
one pair of parallel sides.
Answer:
b
Step-by-step explanation:
plato
I need help with this geometry question asap!
Answers
Answer: True
Explanation: Parallel lines are two lines that lie on the same plane and they never intersect.
To what power must 10 be raised to give 10
Answers
The answer is 1
10¹ = 10
i hope it was helpful
[tex] \sqrt{7} [/tex]
multiplicative inverse (reciprocal)
Answers
Answer: [tex]\frac{\sqrt{7}}{7}[/tex]
========================================================
Explanation:
The reciprocal of x is 1/x where x is nonzero. Multiplying x with 1/x leads to 1. For example, the numbers 9 and 1/9 are multiplicative inverses of each other.
We'll stick 1 over the given square root expression. Then follow these steps to rationalize the denominator.
[tex]\frac{1}{\sqrt{7}}\\\\\\\frac{1*\sqrt{7}}{\sqrt{7}*\sqrt{7}}\\\\\\\frac{\sqrt{7}}{(\sqrt{7})^2}\\\\\\\frac{\sqrt{7}}{7}\\\\[/tex]
In short, [tex]\frac{1}{\sqrt{7}}=\frac{\sqrt{7}}{7}[/tex]
Evaluate bh for b = 12 and h = 2. Type a numerical answer in the space provided.
Answers
The value of bh for b =12 and h = 2 is 24
How to evaluate the expression?The expression is given as:
bh
Where
b = 12 and h = 2
So, we have:
bh = 12 * 2
Evaluate
bh = 24
Hence, the value of bh for b =12 and h = 2 is 24
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a vendor is thinking about changing the number of sweatshirts he brings to tan event. to make sure he doesn't run out, he plans to bring more of the size most likely to be sold. the table shows the number of sweatshirts of each size sold at the vendors last event and the number he had for sale. which size should he bring more of?
Answers
The sweatshirt size which the vendor should bring more of by virtue of that which is most likely to be sold is; Choice A; Extra-large.
Which sweatshirt size should the vendor bring more of?If follows from the task content that the
ratio of extralarge shirts sold relative to the quantity he had to sell is
= 99/108
= 11/12,
This is a ratio which is larger than any other shirt.
Consequently, the vendor must bring more of the X-large shirts to avoid running out of them.
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What is the unit price of a Mt. Dew if a six packs costs $2.70
Answers
Answer:
270 uits
Step-by-step explanation: