Power Function:
Consider the following graphs (1 and 2), and answer the questions FOR EACH GRAPH:
A) In what interval of the graph is it increasing, decreasing and constant? This answer must be justified by means of the definition
B) What is the domain and range?
C) Is it an odd or even function? This answer must be justified by means of the definition
Answers
Graph 1
Part (a)
The function is increasing when x > 0. The function is decreasing when x < 0.
The function is never constant
An increasing portion is when the graph goes uphill when moving left to right. A decreasing portion goes in the opposite direction: it goes downhill when moving left to right.
The reason why the function is never constant is because there aren't any flat horizontal sections. Such sections are when x changes but y does not. No such sections occur.
------------------------
Graph 1
Part (b)
Domain = set of all real numbers
Range = set of y values such that [tex]y \ge 0[/tex]
The domain is the set of all real numbers because we can plug in any value for x without any restriction. There are no division by zero errors to worry about, or square roots of negative numbers to worry about either.
The range is the set of nonnegative numbers as the graph indicates. The lowest y gets is y = 0.
------------------------
Graph 1
Part (c)
The function is even
The function f(x) = 1.6x^12 is an even function due to the even number exponent. For any polynomial, as long as the exponents are all even, then the function itself is even. If all the exponents were odd, then the function would be odd. This applies to polynomials only. A power function is a specific type of polynomial.
Note in the graph, we have y axis symmetry. The mirror line is vertical and placed along the y axis. This is a visual trait of any even function.
We could use algebra to show that f(-x) = f(x) like so
f(x) = 1.6x^12
f(-x) = 1.6(-x)^12
f(-x) = 1.6x^12
The third step is possible because (-x)^12 = x^12 for all real numbers x. It's similar to how (-x)^2 = x^2. You could think of it like (-1)^2 = (1)^2
============================================================
Graph 2
Part (a)
The function is decreasing when x < 0 and when x > 0
The function is never increasing
The function is never constant
In other words, the function is decreasing over the entire domain (see part b). The only time it's not decreasing is when x = 0.
The function is decreasing because the curve is going downhill when moving to the right. You can think of it like a roller coaster of sorts.
At no point of this curve goes uphill when moving to the right. Therefore, it is never increasing. The same idea applies to flat horizontal sections, so there are no constant intervals either.
------------------------
Graph 2
Part (b)
Domain: x is any real number but [tex]x \ne 0[/tex]
Range: y is any real number but [tex]y \ne 0[/tex]
Explanation: If we tried plugging x = 0 into the function, we get a division by zero error. This doesn't happen with any other number. Therefore, the set of allowed inputs is any number but 0.
The range is a similar story. There's no way to get y = 0 as an output.
If we plugged y = 0 into the equation, then we'd get this
y = 17x^(-3)
0 = 17/(x^3)
There's no way to have the right hand side turn into 0. The numerator is 17 and won't change. Only the denominator changes. We can't have the denominator be 0.
------------------------
Graph 2
Part (c)
The function is odd
We can prove this by showing that f(-x) = -f(x)
f(x) = 17x^(-3)
f(-x) = 17(-x)^(-3)
f(-x) = 17* ( -(x)^(-3) )
f(-x) = -17x^(-3)
f(-x) = -f(x)
This is true for nearly all real numbers x, except we can't have x = 0.
Graphic 1:
(A) If f(x) = 1.6x ¹², then f '(x) = 19.2x ¹¹. Both f '(x) and x have the same sign, which means
• for -∞ < x < 0, we have f '(x) < 0, so that f(x) is decreasing on this interval
• for 0 < x < ∞, we have f '(x) > 0, so f(x) is increasing on this interval
f(x) is not constant anywhere on its domain.
(B) Speaking of domain, since f(x) is a polynomial (albeit only one term), it has
• a domain of all real numbers
• a range of {y ∈ ℝ : y = f(x) and y ≥ 0} (in other words, all real numbers y such that y = 1.6x ¹² and y is non-negative)
(C) This function is even, since
f(-x) = 1.6 (-x)¹² = (-1)¹² × 1.6x ¹² = 1.6x ¹² = f(x)
Graphic 2:
(A) Now if f(x) = 17/x ³, then f '(x) = -51/x ⁴. Because x ⁴ ≥ 0 for all x, this means f '(x) < 0 everywhere, except at x = 0. So f(x) is decreasing for (-∞ < x < 0) U (0 < x < ∞).
(B) f(x) has
• a domain of {x ∈ ℝ : x ≠ 0} (or all non-zero real numbers)
• a range of {y ∈ ℝ : y = f(x) and y ≠ 0} (also all non-zero reals)
(C) This function is odd:
f(-x) = 17/(-x)³ = 1/(-1)³ × 17/x ³ = -17/x ³ = -f(x)
Related Questions
Imagine that you are given two linear equations in slope-intercept form. You
notice that both the slopes and the y-intercepts are the same. How many
solutions would you expect for this system of equations?
O A. 1
ОВ. о
C. infinitely many
O D. cannot be determined
SURAT
Answers
Answer:
C. infinitely many
Step-by-step explanation:
If two equations in slope-intercept form have the same slope and y-intercept they must be the same line. Additionally, the solutions of a system of equations are wherever the two lines intersect. Since the lines are the same they must intersect at every point. Therefore, there are infinitely many solutions.
Practice Exercise 3.1 Fill in the blanks: (i) The factors of 12 are (ii) The least non-zero multiples of any number is (iii) ......... is a factor of every number. ing with Numbers
Answers
Answer:
i.)1,2,3,4,6,12
ii).the number itself
iii.)1
Given the recursive formula shown, what are the first 4 terms of the sequence?
Answers
Answer:
5,20,80,320
Step-by-step explanation:
a1 = 5
an = 4 an-1
Let n = 2
a2 = 4 * a1 = 4*5 = 20
Let n = 3
a3 = 4 * a2 = 4*20 = 80
Let n = 4
a4 = 4 * a3 = 4*80 = 320
Sum of × +1 and × + 2
Answers
Step-by-step explanation:
X +1 + X + 2
X + X + 1 + 2
2x + 3
Therefore it's 2x + 3
The product of x and its opposite is always 1.
O True
O False
Answers
Answer:
False
Step-by-step explanation:
x
The opposite of x is -x
x* -x = -x^2
This is not always 1
Answer:
false
Step-by-step explanation:
x*-x= - x ² so its not always 1
so I just started using brainly sorry if its not appropriate
A math professor is wondering if students today are better or worse than in the past. He has given the same final to this year's class that he gave ten years ago. Compute mean, median, and mode for both classes and write a paragraph summarizing the differences.
This Year
35 45 65 75 87
80 69 71 53 90
99 95 70 82 73
93 67 61 57 74
72 77 71 81 83
Ten Years Ago
56 77 75 76 59
74 51 89 55 79
67 77 69 91 68
90 65 79 69 79
87 86 98 91 95
Answers
Answer:
Kindly check explanation
Step-by-step explanation:
Given the following data:
This year :
35, 45, 53, 57, 61, 65, 67, 69, 70, 71, 71, 72, 73, 74, 75, 77, 80, 81, 82, 83, 87, 90, 93, 95, 99
Mean = ΣX / n = 1825 / 25 = 73
The mode = 71 ( most frequently occurring)
Median = 1/2(n+1)th term = 1/(26) = 13th term
Median = 73
10 years ago :
51, 55, 56, 59, 65, 67, 68, 69, 69, 74, 75, 76, 77, 77, 79, 79, 79, 86, 87, 89, 90, 91, 91, 95, 98
Mean = ΣX / n = 1902 / 25 = 76.08
The mode = 79 ( most frequently occurring)
Median = 1/2(n+1)th term = 1/(26) = 13th term
Median = 77
According to the computed statistics, we can conclude that, today is worse than the past as the average score which is almost similar to the median value is higher 10 years ago and the modal score is better 10 years ago as well.
Geometry please help me!In the figure below, what value of x will satisfy the midsegment theorea? X=
Answers
Answer:
x=30.5
Step-by-step explanation:
Using midsegment 's theorea:
[tex]2=\dfrac{RG}{RS} =\dfrac{RH}{RQ} =\dfrac{GH}{SQ} \\\\4x-65=2x-4\\\\2x=61\\\\x=\dfrac{61}{2} \\\\x=30.5\\[/tex]
1. Find the volume of a rectangular block 15 cm long, 5 cm wide and 10 cm length
Answers
9514 1404 393
Answer:
750 cm³
Step-by-step explanation:
The volume is given by the formula ...
V = LWH . . . . where L, W, H represent length, width, height
The volume is the product of the dimensions.
V = (15 cm)(5 cm)(10 cm) = 750 cm³
Which is the same length as 4 kilometers?
Answers
Answer:
A. 4000 meters because
1 km = 1000 meters
and 4 km = 1000 × 4 = 4000
............
Charla has six segments with which to make two triangles. The segments lengths are 2 in., 3 in., 4 in., 5 in., 6 in., and 7 in. Which are possible side lengths of her two triangles?
2 in., 4 in., 6 in. and 3 in., 5 in., 7 in.
2 in., 5 in., 6 in. and 3 in., 4 in., 7 in.
2 in., 3 in., 4 in. and 5 in., 6 in., 7 in.
2 in., 3 in., 6 in. and 4 in., 5 in., 7 in.
Answers
Answer:
The answer is option C
2 in., 3 in., 4 in. and 5 in., 6 in., 7 in.
Step-by-step explanation:
(x - 7)2 = x2 - 49
O True
O False
Answers
Answer:
False
Step-by-step explanation:
A population of deer in Florida grows according to a logistic model, with r = 0.17 and K = 10,000. At what population size is the per capita population growth rate the highest? Group of answer choices N = 1000 N = 5000 N = 8000 N = 10000
Answers
Answer:
N = 1000
Step-by-step explanation:
The population growth of species per capita of any geographical can be computed by using the formula:
[tex]\dfrac{dN}{dT}=rN (1 - \dfrac{N}{K})[/tex]
here;
N = population chance
T = time taken
K = carrying capacity
r = the constant exponential growth rate
From the given equation, we can posit that the value of r will be the greatest at the time the value of dN is highest:
As such, when the population chance = 1000
[tex]\dfrac{dN}{dT}=0.17 * 1000 (1 - \dfrac{1000}{10000})[/tex]
[tex]\dfrac{dN}{dT}=0.17 * 1000 (0.9)[/tex]
[tex]\dfrac{dN}{dT}= 153[/tex]
At N = 5000;
[tex]\dfrac{dN}{dT}= 85[/tex]
At N= 8000;
[tex]\dfrac{dN}{dT}= 34[/tex]
At N = 10000
[tex]\dfrac{dN}{dT}= 0[/tex]
What is the equation of a horizontal line passing through the point (-7,5)?
Oy = 5
Oy = -7
Ox=5
Ox= - 7
Answers
Answer:
1st option, y = 5
Step-by-step explanation:
when the line is horizontal, it's parallel to the x axis
Answer:
y = 5
Step-by-step explanation:
The equation of a horizontal line parallel to the x- axis is
y = c
where c is the value of the y- coordinates the line passes through
The line passes through (- 7, 5 ) with y- coordinate 5 , then
y = 5 ← is the equation of the line
The table shows a linear function.
Which equation represents the function?
x f(x)
-6 -1
-3 4
0 9
3 14
A. f(x)= -5/3x+9
B. f(x)= -5/3x-9
C. f(x)= 9x+5/3
D. f(x)= 5/3x+9
Answers
Answer:
D.
Step-by-step explanation:
Try A:
x = -6, f(x) = -1:-
f(-6) = -5/3(-6) + 9
= 10 + 9 = 19 NOT A.
Try B:
f(-3) = -5/3(-3) - 9
= 5 - 9 = -4 NOT B
Try C:
9(0) + 5/3 = 5/4 NOT C
Try D:
f(3) = 5 + 9 = 14
f(0) = 9, f(-6) = -1 and f(-3) = 4
a grocery store cashier packed 2 carts of groceries equally into 12 paper bags. what fraction of a cart is in each bag?
Answers
Answer:
Step-by-step explanation:
(2 carts)/(12 bags) = (⅙ cart)/bag
GIVING BRAINLIEST!!!!! AND ALL POINTS!!!!!!!!!!!!!!!!!!!
A right rectangular prism is packed with cubes of side length fraction 1 over 4 inch. If the prism is packed with 12 cubes along the length, 8 cubes along the width, and 5 cubes along the height, what is the volume of the prism?
fraction 2 and 3 over 4 cubic inches
fraction 3 and 3 over 4 cubic inches
fraction 7 and 1 over 4 cubic inches
fraction 7 and 1 over 2 cubic inches
Answers
Answer:
7 and 1 over 2 cubic inches ( 7 1/2 in³
Step-by-step explanation:
The height = 1/4 * 5 = 1 1/4 = 1.25
The width = 1/4 * 8 = 2
The length = 1/4 * 12 = 3
Volume = 1.25 * 2 * 3 = 2.5 * 3 = 7.5
0.5 is represented as 1/2
So answer : fraction 7 and 1 over 2 cubic inches or 7 1/2 in³
if my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
A researcher wants to better understand the health benefits of eating vegetables. In a study he finds 300 adults aged 45-60 who eat at least 3 servings of vegetables a day on average. He finds another 200 adults who eat less than 3 servings of vegetables a day on average. The researcher looks at rates of cancer and heart disease in each group and compares both groups. In another study, the researcher finds 500 adults aged 45-60 who eat less than 3 servings of vegetables a day on average, and are willing to participate in a study. The researcher randomly assigns 250 of these adults to a diet which includes 4 servings of vegetables a day. The other 250 continue their usual habits. After 4 years, the rates of cancer and heart disease between the two groups are compared
Identify the statement that correctly states the reason for considering the first study as an observational study and second study as an experiment.
a. In the first study, the treatment is not imposed on the subjects, whereas in the second study the treatment is imposed on the subjects.
b. In the first study, the treatment is not imposed on every subject, whereas in the second study the treatment is imposed on every subject.
c. In the first study, the subjects were not randomly chosen, whereas in the second study the subjects were randomly assigned.
Answers
Answer:
a. In first study, the treatment is not imposed on the subjects, whereas in the second study the treatment is imposed on subjects.
Step-by-step explanation:
In the first study, observation are made on 300 adults who eat 3 servings of vegetables a day on average. The second study has further intensified the research which imposed treatment on the subjects. The random samples of adults are observed in both studies.
37. The trip between 2 towns is exactly 90 miles. You have gone 40% of this distance. How far have
you gone?
Answers
Answer:
36 miles
Step-by-step explanation:
We want to find 40% of 90 miles
40% * 90
.40 * 90
36 miles
We have to find travelled distance inorder to find this we have to find 40℅ of 90miles
[tex]\\ \Large\sf\longmapsto 90\times 40\℅[/tex]
[tex]\\ \Large\sf\longmapsto 90\times \dfrac{40}{100}[/tex]
[tex]\\ \Large\sf\longmapsto 9\times 4[/tex]
[tex]\\ \Large\sf\longmapsto 36miles [/tex]
Find f(-1) given f(x) = –2x^3 + 3x^2 – 22
Answers
[tex]\\ \sf\longmapsto f(-1)[/tex]
[tex]\\ \sf\longmapsto -2x^3+3x^2-22[/tex]
[tex]\\ \sf\longmapsto -2(-1)^3+3(-1)^2-22[/tex]
[tex]\\ \sf\longmapsto -2(-1)+3(1)-22[/tex]
[tex]\\ \sf\longmapsto 2+3-22[/tex]
[tex]\\ \sf\longmapsto 5-22[/tex]
[tex]\\ \sf\longmapsto -17[/tex]
What is the largest value of A according to the division operation given above?
A)300 B)314 C)400 D)450
Answers
Answer:
Hello,
Answer B
Step-by-step explanation:
Since A=15*20+B and B<15
The max for B is 14
==> 300+14=314
PLS HELP I DONT KNOW THIS ONE
Answers
Answer:
x+3
---------------
(x-3)(x-2)(x-4)
Step-by-step explanation:
x+4 x^2 -16
---------------÷ -------------
x^2 - 5x+6 x+3
Copy dot flip
x+4 x+3
--------------- * -------------
x^2 - 5x+6 x^2 -16
Factor
x+4 x+3
--------------- * -------------
(x-3)(x-2) (x-4)(x+4)
Cancel like terms
1 x+3
--------------- * -------------
(x-3)(x-2) (x-4)1
x+3
--------------- x cannot equal 3,2,4 -4
(x-3)(x-2)(x-4)
Need the help thanks guys
Answers
Answer:
x=−5+√29 or x=−5−√29
Step
Let's solve your equation step-by-step.
x2+10x+10=14
Step 1: Subtract 14 from both sides.
x2+10x+10−14=14−14
x2+10x−4=0
For this equation: a=1, b=10, c=-4
1x2+10x+−4=0
Step 2: Use quadratic formula with a=1, b=10, c=-4.
x=
−b±√b2−4ac
2a
x=
−(10)±√(10)2−4(1)(−4)
2(1)
x=
−10±√116
2
x=−5+√29 or x=−5−√29
the x coordinates of the point 2y-x=10 intersect the line yaxis
Answers
Answer:
Point has co-ordinates, (0, 5)
Step-by-step explanation:
If they cut y-axis, then x = 0
[tex]2y - x = 10 \\ 2y - 0 = 10 \\ 2y = 10 \\ y = 5[/tex]
About 12.5% of restaurant bills are incorrect. If 200 bills are selected at ran- dom, find the probability that at least 22 will contain an error. Is this likely or unlikely to occur
Answers
Answer:
0.7734 = 77.34% probability that at least 22 will contain an error. Probability above 50%, which means that this is likely to occur.
Step-by-step explanation:
Binomial probability distribution
Probability of exactly x successes on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
About 12.5% of restaurant bills are incorrect.
This means that [tex]p = 0.125[/tex]
200 bills are selected at random
This means that [tex]n = 200[/tex]
Mean and standard deviation:
[tex]\mu = E(X) = np = 200*0.125 = 25[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{200*0.125*0.875} = 4.677[/tex]
Find the probability that at least 22 will contain an error.
Using continuity correction, this is [tex]P(X \geq 22 - 0.5) = P(X \geq 21.5)[/tex], which is 1 subtracted by the p-value of Z when X = 21.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{21.5 - 25}{4.677}[/tex]
[tex]Z = -0.75[/tex]
[tex]Z = -0.75[/tex] has a p-value of 0.2266.
1 - 0.2266 = 0.7734
0.7734 = 77.34% probability that at least 22 will contain an error. Probability above 50%, which means that this is likely to occur.
which of the following are ordered pairs for the given function f(x)=1+x.? (1,2) (3,3) (0,2) (1,0) (0,1)
Answers
Answer:
no,
(
1
,
0
)
is not an ordered pair of the function
f
(
x
)
=
1
+
x
.
Step-by-step explanation:
Ordered pairs are usually written in the form
(
x
,
y
)
by tradition.
so usingthe function,
f
(
x
)
=
1
+
x
we can rewrite it as,
y
=
1
+
x
any pair of x and y that satisfy this equation are solutions to the equation.
so subbing in
(
1
,
0
)
,
0
=
1
+
(
1
)
0
=
2
which is not true so the point does not make the function true.
It might be easier to see graphically,
graph{1+x [-10, 10, -5, 5]}
any combination of x and y on this line make the equation true and as such are an ordered pair of the function.
Answer:
Step-by-step explanation:
a test for diabetes results in a positive test in 95% of the cases where the disease is present and a negative test in 07% of the cases where the disease is absent. if 10% of the population has diabetes, what is the probability that a randomly selected person has diabetes, given that his test is positive
Answers
Answer:
0.9378 = 93.78% probability that a randomly selected person has diabetes, given that his test is positive.
Step-by-step explanation:
Conditional Probability
We use the conditional probability formula to solve this question. It is
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which
P(B|A) is the probability of event B happening, given that A happened.
[tex]P(A \cap B)[/tex] is the probability of both A and B happening.
P(A) is the probability of A happening.
In this question:
Event A: Positive test
Event B: Person has diabetes.
Probability of a positive test:
0.95 out of 0.1(person has diabetes).
0.007 out of 1 - 0.1 = 0.9(person does not has diabetes). So
[tex]P(A) = 0.95*0.1 + 0.007*0.9 = 0.1013[/tex]
Probability of a positive test and having diabetes:
0.95 out of 0.1. So
[tex]P(A \cap B) = 0.95*0.1 = 0.095[/tex]
What is the probability that a randomly selected person has diabetes, given that his test is positive?
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.095}{0.1013} = 0.9378[/tex]
0.9378 = 93.78% probability that a randomly selected person has diabetes, given that his test is positive.
Solve for x. Round to the nearest tenth, if necessary.
S
540
R
2.3
X
O
Please help
Answers
Answer:
[tex]\boxed {\boxed {\sf x \approx 1.9}}[/tex]
Step-by-step explanation:
We are asked to find x, a missing side in a triangle.
This is a right triangle because there is a small square in the corner representing a 90 degree or right angle. Therefore, we can use right triangle trigonometry. The three main functions are:
sinθ= opposite/hypotenuse cosθ= adjacent/hypotenuse tanθ= opposite/adjacentExamine the triangle. We will use angle S, measuring 54 degrees, for theta. Side QR, measuring x, is opposite angle S. Side QS, measuring 2.3, is the hypotenuse because it is opposite the right angle. Since we have the opposite and hypotenuse, we will use sine.
[tex]sin \theta = \frac {opposite}{hypotenuse}[/tex]
θ= 54opposite= x hypotenuse = 2.3[tex]sin (54)= \frac{ x}{2.3}[/tex]
We are solving for x, so we must isolate the variable. It is being divided by 2.3 The inverse operation of division is multiplication, so we multiply both sides by 2.3
[tex]2.3* sin (54)= \frac{x}{2.3}*2.3[/tex]
[tex]2.3* sin (54)=x[/tex]
[tex]2.3*0.8090169944=x[/tex]
[tex]1.860739087 =x[/tex]
Round to the nearest tenth. The 6 in the hundredth place to the right tells us to round the 8 up to a 9.
[tex]1.9 \approx x[/tex]
x is approximately 1.9
Last year, Rob set up the Road Runner Race for his school.
The race was 1,200 meters long and 188 people signed up to
run the race. 38 people did not show up to run. This year,
there will be 3 times as many runners as last year. How
many people will run the race this year?
Answers
Answer:
450 runners
Step-by-step explanation:
i need help and thx you freinds
Answers
Answer:
Below
Step-by-step explanation:
Find the areas of the triangles on the sides
A = bh / 2
= (3)(5) / 2
= 7.5
There are 2 of these so it would just be 15
Now for the square
A = lw
= (5)(6)
= 30
Add em all up
Total area = 15 + 30
= 45 cm^2
Hope this helps!
The scores of a high school entrance exam are approximately normally distributed with a given mean Mu = 82.4 and standard deviation Sigma = 3.3. What percentage of the scores are between 75.8 and 89?
Answers
Notice that
75.8 = 82.4 - 6.6 = 82.4 - 2 × 3.3
89 = 82.4 + 6.6 = 82.4 + 2 × 3.3
Then the percentage of students with scores between 75.8 and 89 make up the part of the distribution that lies within 2 standard deviations of the mean. The empirical (68-95-99.7) rule says that approximately 95% of any distribution lies within this range.
Answer:
b
Step-by-step explanation:
