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.
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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.
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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.
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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
The graph of the equation y = 3x + 1 is shown below.
If the graph is reflected across the y-axis, what will be the equation of the new graph?
Answers
Answer:
I think its B. Y = -3x + 1
Step-by-step explanation:
Sorry if this is wrong.
Assume x and y are two odd numbers and x/y is an integer.
Which of the following statements are true?
I. x + y is odd
2. xy is odd.
3. x/y is odd
4. x-y is odd
Answers
Answer:
Let us check these out one at a time:
1. x + y is odd. FALSE. The sum of 2 odd numbers is even.
2. xy is odd. TRUE. The product of 2 odd numbers is odd.
3. x/y is odd. TRUE. The ratio of 2 odd numbers is odd, if the ratio is an integer.
4. x - y is odd. FALSE. The difference of 2 odd numbers is even.
Only statements 2 and 3 are TRUE, so that makes (C) the correct answer.
Function: y = x^2+ 5x - 7
Vertex:(
Solutions:(
and
Answers
Answer:
Vertex: [tex](-\frac{5}{2} , -\frac{53}{4})[/tex]
Solutions: (0,-7), (1, -1), (2, 7)
I hope this helps!
Digits to write five hundred seven billion,six hundred forty million,seven hundred forty-two thousand,seventy two
Answers
Pete receives a weekly allowance for doing chores around the
house. Pete saves his money for 17 weeks. After 17 weeks, he
saves $187. How much money does Pete get for his allowance
each week
Answers
9514 1404 393
Answer:
A. $11 per week
Step-by-step explanation:
$187 = (17)(weekly allowance) . . . . . Pete's savings after 17 weeks
weekly allowance = $187/17 = $11 . . . . . divide by 17
Pete gets $11 each week.
Answer:11 per week
Step-by-step explanation:I did the test
find the solution to the system of equations.
y= -7x + 3
y= -x - 3
Answers
Answer:
x = 1 y = -4
Step-by-step explanation:
-7x + 3 = -x - 3
-7x = -x - 6
-6x = -6
x = 1
y = - (1) - 3
y = -1 - 3
y = -4
find the difference between the product of 2.5 and 7.5 and the sum of2.75 and 9.55
Answers
Answer:
6.45
Step-by-step explanation:
The answer=2.5*7.5-(2.75+9.55)=18.75-(12.3)=6.45
The table and the Circle Chart shown display the percentage of dogs in seven different groups of dog breeds in a dog competition.
How does the Circle Graph misrepresent the data in the table?
A. The percentages do not add up to 100.
B. The size of the section representing the Non-Sporting group is smaller than the section that represents Herding group.
C. The number of regions in the graph is not correct for the data.
D. One of the breeds in the table is not represented in the Circle Chart.
Answers
Answer:
a
Step-by-step explanation:
Answer:
Incorrect, it's actually B.
Step-by-step explanation:
Look at the Non-Sporting Group and Herding Group. The sizes of the slices are different (NSG being smaller than HG)
pls help me asap !!!!!!
Answers
9514 1404 393
Answer:
(-2.5, 1)
Step-by-step explanation:
The desired point is the weighted average of the end points, where the weight of each point is the fraction of distance from the opposite end.
P = 5/8(-1, 7) +3/8(-5, -9) = (-5-15, 35-27)/8 = (-2.5, 1)
The desired point is (-2.5, 1).
what is this expression in simplest form? x^2+x-2/x^3-x^2+2x-2
Answers
The Required expression is (x+2) / (x^2+2).
x^2+x-2/x^3-x^2+2x-2
The process in mathematics to operate and interpret the function to make function simple or more understandable called simplify and the process is called simplification.
= (x^2+x-2) / (x^3-x^2+2x-2)
= (x-1)(x+2) / (x^2+2)(x-1)
=(x+2) / (x^2+2)
Thus, The Required expression is (x+2) / (x^2+2).
Learn more about simplification here:
https://brainly.com/question/12501526
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Answer: see photo#2
Step-by-step explanation:
Two-thirds of the difference of 4 and .x.
Answers
Answer:
DIFFERENCE OF 4 AND x => 4 - x
two thirds so final expression is
[tex]\frac{2}{3} (4-x)[/tex]
HELPPPPP ASP PLZZZZZ
Answers
Answer:
[tex](f-g)(x)[/tex]
[tex]f(x)-g(x)[/tex]
[tex]x^{2} -6x-27-x+9[/tex]
[tex]x^{2} -7x-18[/tex]
----------------------
[tex](f*g)(x)[/tex]
[tex]=f(x)g(x)[/tex]
[tex](x^{2} -6x-27)(x-9)[/tex]
[tex]=x^{3} -15x^{2}+27x+243[/tex]
----------------------
[tex]\frac{f}{g} (x)[/tex]
[tex]\frac{x^{2} -6x-27}{x-9}[/tex]
[tex]\frac{(x-9)(x+3)}{x-9}[/tex]
[tex]x+3[/tex]
-----------------------
[tex](f+g)(x)[/tex]
[tex]f(x)+g(x)[/tex]
[tex]=x^{2} -6x-27+x-9[/tex]
[tex]=x^{2} -5x-36[/tex]
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OAmalOHopeO
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Any help is appreciated.
No links pls
Answers
Answer:
its b plz give brainlist
Step-by-step explanation:
Martina has 240 meters of fencing and wishes to form three sides of a rectangular field. The fourth side borders a river and will not need fencing.
As shown below, one of the sides has length x (in meters).
x
Side along river
(a) Find a function that gives the area Ax of the field (in square meters) in terms of x.
=Ax
(b) What side length x gives the maximum area that the field can have?
Side lengthx:meters
(c) What is the maximum area that the field can have?
Maximum area:square meters
Answers
Answer:
Step-by-step explanation:
Answering a comes from simplification, and answering b and c are done all in one step: completing the square on the quadratic that results from a.
(a) If Martina has 240 m of fencing and is only utilizing one side for the length and 2 sides for the width, the perimeter formula is
240 = x + 2w where x is a length and w is the width. Solving this for w in terms of x:
240 - x = 2w so
[tex]w=120-.5x[/tex] The area for a rectangle is L * W, so our area using the lengths we have is
A(x) = x(120 - .5x) and we simplify:
A(x) = 120x - .5x² That's the answer to a.
Now for b and c, we will complete the square on this to get the vertex.
Begin by factoring out the -.5:
[tex]A(x)=-.5(x^2-240x)[/tex] Now we take half the linear term, square it and add it both inside the parenthesis and outside the parenthesis. Our linear term is 240. Half of 240 is 120, and 120 squared is 14400:
[tex]A(x)=-.5(x^2-240x+14400)+7200[/tex] (The 7200 comes from multiplying the 14400 times the -.5; -.5 times 14400 is -7200 so to balance things out, we have to add 7200).
The perfect square binomial that results from this is
A(x) = -.5(x - 120)² + 7200. From this we determine that our vertex is
(120, 7200). The 120 is the value of x, the length we are asked to find in b; the 7200 is the max area we are asked to find in c.
The required solutions are,,
(a) area = 240x - 2x²
(b) the side adjacent to the rivers gives the maximum length of the field.
(c) the maximum area could be 6400-meter square.
The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
length of the field is x,
The perimeter of the field, = 240
x + x + width = 240
width = 240 - 2x
now,
(a)
area of the field,
= length * width,
= x(240-2x)
= 240x - 2x²
Similarly,
(b) the side adjacent to the rivers gives the maximum area of the field.
(c) the maximum area could be 6400-meter square.
Thus, the required solutions are mentioned above.
Learn more about simplification here: https://brainly.com/question/12501526
#SPJ2
Find a fraction equivalent to 2/3 that has a denominator of 6. Then add the fraction to 1/6. What is the sum?
THIS IS FROM SAXON MATH
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Answers
5/6
Step-by-step explanation:
2/3 is equal to 4/6 and if you add 1/6 + 4/6 you get 5/6
Answer:
Step-by-step explanation:
4/6 is equivalent to 2/3 (Times by two).
Since we now have a fraction with the denominator of 6, we can add 1/6 easily.
4/6 + 1/6 = 5/6
The sum is 5/6
write each equation explicitly in terms of x. then indicate whether the equation is a function.
y^2-x^2+1=50
Answers
Answer:
Hello,
Step-by-step explanation:
[tex]y^2-x^2+1=50\\y^2=x^2+49\\2\ functions \ :\\\\y=\sqrt{x^2+49} \\\\or\\\\y=-\sqrt{x^2+49} \\[/tex]
Using function concepts, it is found that:
The explicit equation in terms of x is given by: [tex]y = \pm \sqrt{x^2 + 49}[/tex]The equation is not a function, as there are multiple outputs for a single input.----------------------
The expression is given by:
[tex]y^2 - x^2 + 1 = 50[/tex]
In terms of x, the equation is given by:
[tex]y^2 = 50 + x^2 - 1[/tex]
[tex]y^2 = x^2 + 49[/tex]
[tex]y = \pm \sqrt{x^2 + 49}[/tex]
----------------------
An equation is a function if for each value of the input, there is only one output.Testing the input x = 0:
[tex]y = \pm \sqrt{0^2 + 49}[/tex]
[tex]y = \pm \sqrt{49}[/tex]
[tex]y = \pm 7[/tex]
Two output values for one input, thus, it is not a function.
A similar problem is given at https://brainly.com/question/24603090
I will give brainly.
How do you determine if a slope is positive or negative?
Answers
You have to find the slope .
How?
Take 2points
(x1,y1)(x2,y2)Slope formula[tex]\\ \rm\Rrightarrow \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Step-by-step explanation:
What the Slope Means A positive slope means that two variables are positively related—that is, when x increases, so does y, and when x decreases, y also decreases. Graphically, a positive slope means that as a line on the line graph moves from left to right, the line rises.
Can someone please help me with this math problem
Answers
We have [tex]f\left(f^{-1}(x)\right) = x[/tex] for inverse functions [tex]f(x)[/tex] and [tex]f^{-1}(x)[/tex]. Then if [tex]f(x) = 2x+5[/tex], we have
[tex]f\left(f^{-1}(x)\right) = 2f^{-1}(x) + 5 = x \implies f^{-1}(x) = \dfrac{x-5}2[/tex]
Then
[tex]f^{-1}(8) = \dfrac{8-5}2 = \boxed{\dfrac32}[/tex]
Help pls and thank you :)
Answers
Answer: B
[tex]tan(N)=\frac{opposite}{adjacent}=\frac{n}{r}[/tex]
A survey is sent out to 500 people asking questions related to the link between religion and sexual behavior at a young age. The survey does not have any tracking device, and the results are sent to a database that gives no indication of who or where the results came from. However, the database is posted online for religion and human behavior researchers to freely conduct statistical analyses. The responses to this survey are
a. both confidential and anonymous.
b. confidential only.
c. anonymous only.
d. neither confidential nor anonymous.
Answers
Answer:
c
Step-by-step explanation:
Since the the identities of the survey takers were kept secret, it would be anonymous. If it wasn't anonymous, then people would know who took the survey.
Since the survey results were released online for analyses, it is not confidential. If it were to be confidential, then the survey distributors would have kept the results to themseles.
(g o f)(6)
A. Find f(6)
B. Substitute the value you found in part A into g(x) to find g(f(6))
Answers
Step-by-step explanation:
A. gof=g(f(x))
= g(f(6))
=6×6
36
6x2 - 7x - 5 =0
Let x = j and x = k be solutions to the equation above, with j > k. What is the value of j - k?
Answers
9514 1404 393
Answer:
j-k = 13/6
Step-by-step explanation:
The quadratic formula tells you the two solutions are ...
x = (-b/(2a)) ±√(b² -4ac)/(2a)
The difference between these solutions is ...
j-k = √(b² -4ac)/a
j-k = √((-7)² -4(6)(-5))/6 = √(49 +120)/6 = √169/6
j-k = 13/6
• What is the constant term in the expression 3x + 11?
Answers
Answer:
11
Step-by-step explanation:
A constant term in an expression or equation contains no variables. In other words, it’s just number on its own. For example: f (x) = 2x2 + 3 (the constant term is 3). Other examples of constant terms: 5, -99, 1.2 and pi (π = 3.14…).
Answer: Constant = 11
Concept:
In an algebraic expression, there are three main kinds of terms:
Constants: an individual numberVariable: the unknown valueCoefficient: the number before a variableIf you are still confused, you may refer to the attachment below for a graphical explanation.
Solve:
Given expression: 3x + 11
As we know, the constant is defined as an individual number. Thus, as we can see from the given expression, 11 is the only number that is isolated and being an individual.
Hope this helps!! :)
Please let me know if you have any questions
A scientist is studying the growth and development of an epidemic virus with a growth rate of 9% per month that has infected 3,124 people. If this rate continues, what will be the number of infected people in another 9 months? Round your answer to the nearest whole number.
Answers
Answer:
About 6,785 people will be infected about nine months.
Step-by-step explanation:
We can write an exponential function to represent the situation. The standard exponential function is given by:
[tex]\displaystyle f(x) = a(r)^x[/tex]
Where a is the initial value, r is the rate, and x, in this case, is the time that has passed in months.
3,124 people have already been infected. Thus, our initial value a = 3124.
And an additional 9% will be infected per month. Therefore, our rate r will be 1 + 9% or 1.09.
Hence, our function is:
[tex]\displaystyle f(x) = 3124(1.09)^x[/tex]
Then after nine months, the total amount of infected people will be f(9):
[tex]\displaystyle f(9) = 3124(1.09)^{(9)}[/tex]
Use a calculator:
[tex]\displaystyle f(9) \approx 6785[/tex]
About 6,785 people will be infected about nine months.
Answer:
7,022
Step-by-step explanation:
What is the inverse of the function a(x)=1/x-2
Answers
Answer:
x = 1/x - 2
Step-by-step explanation:
Express b+1/3b-2 with “b” as the subject
Answers
Answer:
b = [tex]\frac{1+2a}{3a-1}[/tex]
Step-by-step explanation:
Given
a = [tex]\frac{b+1}{3b-2}[/tex] ( multiply both sides by 3b - 2 )
a(3b - 2) = b + 1 ← distribute left side
3ab - 2a = b + 1 ( subtract b from both sides )
3ab - b - 2a = 1 ( add 2a to both sides )
3ab - b = 1 + 2a ← factor out b from each term on the left side
b(3a - 1) = 1 + 2a ( divide both sides by 3a - 1 )
b = [tex]\frac{1+2a}{3a-1}[/tex]
Answer:
[tex]→a = \frac{(b + 1)}{(3b - 2)} \\ a(3b - 2) = (b + 1) \\ 3ab - 2a = b + 1 \\ 3ab - b = 2a + 1 \\ b(3a - 1) =( 2a + 1) \\ \boxed{b = \frac{(2a + 1)}{(3a - 1)} }✓[/tex]
b = (2a+1)/(3a-1) is the right answer.Functions, f and g are given by f(x)= 3+ cos x and g(x) = 2x, x is a real number. Determine the value of c for which f(g(x))= g(f(x)) where 0[tex]\leq[/tex] x<2[tex]\pi[/tex]
Answers
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Answer:
x = π
Step-by-step explanation:
You want f(g(x)) = g(f(x)):
3 +cos(2x) = 2(3 +cos(x))
cos(2x) -2cos(x) = 3 . . . . . . . rearrange
2cos(x)²-1 -2cos(x) = 3 . . . . . use an identity for cos(2x)
2(c² -c -2) = 0 . . . . . . . . . . . . substitute c = cos(x)
(c -2)(c +1) = 0 . . . . . . . . . . . factor
c = 2 (not possible)
c = -1 = cos(x) . . . . . true for x = π
The value of x that makes f(g(x)) = g(f(x)) is x = π.
_____
Additional comment
The substitution c=cos(x) just makes the equation easier to write and the form of it easier to see. There is really no other reason for making any sort of substitution. In the end, the equation is quadratic in cos(x), so is solved by any of the usual methods of solving quadratics.
If x and x+30 are the pair of co - interior angle
Answers
answer in screenshot
Select the correct answer.
Select the function that defines the given sequence.
-8, -20, -50, –125, 625,...
ОА
- 1)
f(n) = -8-()
n-
OB.
- 1)
= -8.(29)
OC f(1) = -8
f(n) = f(n − 1), for n = 2, 3, 4, ...
OD. (1) = 8
f(n) = - . 1(n − 1), for n = 2, 3, 4, ...
Answers
Answer:
Option C
Step-by-step explanation:
f(1) = -8
f(2) = -5/2×-8 = -20
f(3) = -5/2×-20 = -125
.
.
.
So option C is the answer
Answer:
C. a1 = -8
an = an-1 * 5/2 for n >1
Step-by-step explanation:
-8, -20, -50, –125, 625,...
We need to find the common ratio
Take the second term and divide by the first term
-20/-8 = 5/2
Take the third term and divide by the second term
-50/-20 = 5/2
The common ratio is 5/2
A geometric sequence is
an = a1 * r^(n-1)
an = -8 *(5/2)^(n-1)
We can also write a recursive formula
a1 = -8
an = an-1 * 5/2 for n >1
Help me with this please
Answers
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Answer:
B. √6
Step-by-step explanation:
The circles are not tangent to one another. If they were, the distance between their centers would be the sum of their radii: 1 +1 = 2.
__
The center of the first circle is (√3, √3), and the center of the second is the origin. The distance between these two centers is given by the distance formula:
d = √((x2 -x1)^2 +(y2 -y1)^2)
d = √((√3 -0)^2 +(√3 -0)^2) = √(3+3) = √6 . . . . matches choice B
