Answers
(1) f(x) = (1 - x³) / (x - 1)
(a) The domain is the set of values that this function can take on. If x = 1, the denominator becomes 0 and the function is undefined. Any other value of x is okay, though, since for x ≠ 1, we have
f(x) = (1 - x³) / (x - 1) = - (1 - x³) / (1 - x) = -(x² + x + 1)
which is defined for all x. This also tells us that the plot of f(x) is a parabola with a hole at x = 1. So, the domain is the interval (-∞, 1) ∪ (1, ∞).
(b) The range is the set of values that the function actually does take on. Taking the simplified version of f(x), we can complete the square to write
-(x² + x + 1) = -(x² + x + 1/4 - 1/4) - 1 = -(x + 1/2)² - 3/4
which is represented by a parabola that opens downward, with a maximum value of -3/4. So the range is the interval (-∞, -3/4).
(c) Judging by the plot of f, the limits at both negative and positive infinity are -∞.
(d) Same answer as part (a).
(2) f(x) = x³ - x
(a) The derivative of f at x = 3, and hence the slope of the tangent line to this point, is
[tex]f'(3)=\displaystyle\lim_{h\to0}\frac{f(3+h)-f(3)}h[/tex]
[tex]f'(3)\displaystyle=\lim_{h\to0}\frac{((3+h)^3-(3+h))-24}h[/tex]
[tex]f'(3)=\displaystyle\lim_{h\to0}\frac{(27+27h+9h^2+h^3)-(3+h)-24}h[/tex]
[tex]f'(3)=\displaystyle\lim_{h\to0}\frac{26h+9h^2+h^3}h[/tex]
[tex]f'(3)=\displaystyle\lim_{h\to0}(26+9h+h^2)=\boxed{26}[/tex]
(b) The tangent line at x = 3 has equation
y - f (3) = f ' (3) (x - 3)
y - 24 = 26 (x - 3)
y = 26 x - 54
We also want to find any other tangent lines parallel to this one, which requires finding all x for which f '(x) = 26. We could use the same limit definition as in part (a), but to save time, we exploit the power rule to get
f '(x) = 3 x² - 1
Then solve for when this is equal to 26:
3 x² - 1 = 26 ==> x² = 9 ==> x = ±3
The other tangent line occurs at x = -3, for which we have f (-3) = -24, and so the equation for the tangent is
y - f (-3) = 26 (x - (-3))
y + 24 = 26 (x + 3)
y = 26 x + 54
Related Questions
the product of 9 and the sum of a number z and 4
Answers
Place the terms into the correct column that describes the reactions of photosynthesis.
Word bank:
Processes: ATP Synthase, Calvin Cycle, Electron Transport Chain, Photosystem I, Photosystem II
Locations: Stroma, Thylakoids membrane
Reactants: ADP, ATP, Carbon Dioxide, NADP+, NADPH, Sunlight, Water
Products: ADP, ATP, NADPH, NADP+, Oxygen, Sugars
Answers
Answer:
Please find the correct placement in a tabular format in the attached image. The explanation has been done below.
Step-by-step explanation:
The process of photosynthesis occurs in two stages viz: light-dependent and light-independent reactions.
- In the light-dependent reaction, which occurs in the THYLAKOID MEMBRANE of the chloroplast, energy from SUNLIGHT is used to breakdown WATER to release electrons in order to synthesize ATP and NADPH from ADP and NADP+. In a nutshell, the processes involved in this stage are Electron transport chain, photosystem I, photosystem II, and ATP synthase.
- In the light-independent stage, also called CALVIN CYCLE, the ATP, NADPH, and CO2 are used as reactants to synthesize SUGAR (glucose), NADP+ and ADP (which goes back to the first stage) as products.
Solve for x. 2x + 5 = 3x -1
Answers
Answer:
x = 6
Step-by-step explanation:
Isolate the variable by dividing each side by factors that don't contain the variable.
6 = 1x
x = 6
The value of x is 6
Diana goes on vacation with $260. Each day, she spends the same amount of
money. After 1, 2, 3, and 4 days on vacation, she has $242, $224, $206, and
$188, respectively.
Please help
Answers
I need help with this question
Answers
Answer:
49 or 1
Step-by-step explanation:
answer 1 14+10+13+12=49 answer 2 (14+10)-(13+12)=1
During each week of piano lessons, Deb learns to play 2 new pieces. Inall, how many weeks of piano lessons will Deb need before she will be able to play a total of 26 pieces?
_______Weeks
Answers
Answer:
13 weeks
Step-by-step explanation:
it will take 13 weeks because 26 devided by 2 is 13.
The radius of a circle is 10 m. Find its area in terms of 7.
Answers
Answer:
[tex]\pi \: r{2} \\ = \frac{22}{7} \times 10 {}^{2} \\ = 3.14 \times 100 \\ = 314[/tex]
I think the Answer is 314.....
Thank you ☺️
Find the measure of the missing angle
Answers
Consider the function and its inverse.
and
When comparing the functions using the equations, which conclusion can be made?
f(x) = x^2 +4 and f^-1 (x) = square root of x-4
The domain of f(x) is restricted to x ≥ 0, and the domain of f–1(x) is restricted to x ≥ 0.
The domain of f(x) is restricted to x ≥ 0, and the domain of f–1(x) is restricted to x ≤ 0.
The domain of f(x) is restricted to x ≤ 0, and the domain of f–1(x) is restricted to x ≥ 4.
The domain of f(x) is restricted to x ≤ 0, and the domain of f–1(x) is restricted to x ≤ 4.
Answers
Answer:
The domain of [tex]f(x)[/tex] is restricted to [tex]x\leq 0[/tex], and the domain of [tex]f^{-1}(x)[/tex] is restricted to [tex]x\geq 4[/tex].
Step-by-step explanation:
From Function Theory we know that domain of a function is the set of values such that an image exist. Let [tex]f(x) = x^{2}+4[/tex] and [tex]f^{-1}(x) = \sqrt{x-4}[/tex] the function and its inverse, respectively.
At first glance we notice that function is a second order polynomial and every polinomial is a continuous function and therefore, there exists an image for every element of domain.
But domain of its inverse is restricted to every value of x so that [tex]x-4 \geq 0[/tex], which means that [tex]x\geq 4[/tex].
Finally, we concluded that following answer offers the best approximation to our result:
The domain of [tex]f(x)[/tex] is restricted to [tex]x\leq 0[/tex], and the domain of [tex]f^{-1}(x)[/tex] is restricted to [tex]x\geq 4[/tex].
Answer:
C. The domain of f(x) is restricted to x ≤ 0, and the domain of f–1(x) is restricted to x ≥ 4.
Step-by-step explanation:
g÷2+ 19 > 15
what is g?
Answers
Answer: g > -8
Step-by-step explanation:
sadmep
which one doesn’t belong? please help asap
Answers
Answer:
The one without an angle
Step-by-step explanation:
I'm guessing because the others ones all have angle measures (30, 22, x) while the bottom left doesn't.
A nurse has an annual income of $52,175. The income tax the nurse must pay is 9%. What is the amount of income tax in dollars and cents that the nurse must pay?
Answers
Answer:
The amount of income tax in dollars and cents is $4,695.75
Step-by-step explanation:
To find the amount of income tax that should be paid, we simply use the percentage said to be for the income tax
Mathematically;
income tax = 9% of 52,175
= 9/100 * 52175
= $4695.75
The income tax to be paid on the income is $52,175
Answer:4695.75
Step-by-step explanation:
Well if u made 52175 a whole then put 9 as a percentage you would cross multiply 52175 with 9 then divided with the whole percentage (100)
You believe that requiring students to attend a series of presentations on study skills will improve their grades. You are concerned that boys and girls might react differently to such a series, and that students in grades 9 and 10 might react differently than those in grades 11 and 12. You set up an experiment involving 100 students in ninth and tenth grade and 100 students in eleventh and twelfth grade. (Assume there are equal numbers of girls and boys in each grade.) You intend to measure improvement based on pre- and post-treatment grade-point averages. A. Describe the design of an experiment to help you determine if a set of presentations on study skills is effective in improving grades. AP Statistics Page 5 of 13 Exam: Final Exam User Name: Instructor: Date: (print clearly) B. Suppose you have 50 students and need to assign them to two groups of equal size. Describe a randomization procedure that would achieve this. C. Suppose you have 50 students and need to assign them to two independent groups. Describe a randomization procedure that would achieve this.
Answers
Answer:
it will be equal and keep it up you will make good grades
Step-by-step explanation:
The design of an experiment in the research to help determine if a set of presentations is simply to consider two groups depending on the gender.
What is a research?It should be noted that a research simply means an enquiry that is done to achieve a particular information.
In this case, the design of an experiment in the research to help determine if a set of presentations is simply to consider two groups depending on the gender.
Also, the randomization procedure that would achieve this is the ANOVA.
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will give u brainliest if i get it write hurry!!!
Answers
Answer:
c trust me
and well i just did that 20 mins ago
Answer:
C for sure hope you get it
For each of the following variables, determine whether the variable is categorical or numerical. If the variable is numerical, determine whether the variable is discrete or continuous. a. Amount of time spent shopping for clothing in the past week b. Preferred brand of shoes c. Number of deliveries received at the household per week d. Most likely time period during which shopping for clothing takes place (weekday, weeknight, or weekend) a. Amount of time spent shopping for clothing in the past week is what kind of variable? categorical continuous numerical discrete numerical b. Preferred brand of shoes is what kind of variable? categorical discrete numerical continuous numerical c. Number of deliveries received at the household per week is what kind of variable? continuous numerical discrete numerical categorical d. Most likely time period during which shopping for clothing takes place is what kind of variable? continuous numerical discrete numerical categorical
Answers
Answer:
a. Amount of time spent shopping for clothing in the past week = CONTINUOUS NUMERICAL
b. Preferred brand of shoes = CATEGORICAL
c. Number of deliveries received at the household per week = DISCRETE NUMERICAL
d. Most likely time period during which shopping for clothing takes place (weekday, weeknight, or weekend) = CATEGORICAL
Step-by-step explanation:
Determine whether variable is Categorical or Numerical (Discrete or continuous) :
Categorical variables are used in grouping data values into a defined set of classes or distinct groups. Numerical variables are those usually represented with numbers and this posses numerical meaning. Numerical variables can either be discrete when the values in the dataset are finite or continuous, when values aren't finite and usually uncountable.
a. Amount of time spent shopping for clothing in the past week = CONTINUOUS NUMERICAL
b. Preferred brand of shoes = CATEGORICAL
c. Number of deliveries received at the household per week = DISCRETE NUMERICAL
d. Most likely time period during which shopping for clothing takes place (weekday, weeknight, or weekend) = CATEGORICAL
Categorical Variables : b) Preferred shoe brand; & d) Likely period of shopping
Numerical Continuous Variable : a) Amount of shopping time ; & Numerical Discrete Variable : c) Number of deliveries
Numerical Variables, as the name denotes - can be expressed in numbers. Like : Number of variable x.
Discrete Variables are only in real values : like 1,2,3 etc. Continuous Variables can take values between these real values also : like 1.4 , 1.5 , 2.8 etc. Example : Number of People is discrete, Height & Weight can be continuous.Categorical Variables, as the name denotes - cant be expressed n numbers but rather in categories. Like : A person being man, woman or other gender.
A] Amount of time spent shopping for clothing in the past week : Numerical & Continuous, can be in decimal values like 2.8 , 4.5 hours etc
B] Preferred brand of shoes : Categorical Variable, will be amongst the brand names like Pima, Nuke etc
C] Number of deliveries received at the household per week : Numerical & Discrete , can be only in non decimal whole values - like 4 , 5 , 10 etc
D] Most likely time period during which shopping for clothing takes place : Categorical Variable, will be amongst the mentioned period of weeks or days - like weekday, weeknight, or weekend
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HELP ASAP IT'S AN EMERGENCY
Answers
Answer:
AAS
Step-by-step explanation:
Two angles are equivalent and they share a side
At a bake sale there were 72 raisin cookies sold. If the ratio of raisin cookies sold to oatmeal cookies sold was 9 : 1, what is the combined amount of raisin and oatmeal cookies sold?
Answers
Answer:
80
Step-by-step explanation:
72/9 is 8
9x8 is 72
so 1x8 is 8
72 plus 8 is 80
What do the following two equations represent? 4x-2y= -5 and y=-1/2x-1/10
Answers
Answer:
perpendicular lines
Step-by-step explanation:
Kahn Academy
The given lines are intersecting but not perpendicular.
What is intersecting lines?In Euclidean geometry, the empty set, a point, or another line can intersect another line. For instance, computer graphics, motion planning, and collision detection can all benefit from differentiating these cases and locating the intersection.
Given:
y = 6x - 2
2x - 12y = 24
⇒ 6x - y = 2 ..(1)
2x - 12y = 24 ..(2)
Multiply second equation by 3,
6x - 36y = 72 ..(3)
Subtract equation (1) from equation (3).
6x - 12y = 24
- 6x - y = 2
_______________
-11y = 22
y = -2
Plug y = -2 in equation (1)
6x - (-2) = 2
6x + 2 = 2
6x = 2 - 2
6x = 0
x = 0
The solution of the given system is (0, -2).
Hence, the given lines are intersecting but not perpendicular.
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complete question:
What do the following two equations represent?
.y=6x-2
.2x-12y=24
Use 4's to create problems that will equal 1-12. Remember to use the correct order of operations to solve your problems: Parentheses, Exponents, Multiply or Divide, Add or Subtract.
Answers
Answer and Step-by-step explanation:
Solution:
Order of operations:
1- Parentheses
2- Exponent
3- Multiply
4- Divide
5- Subtract
By using these operations we can find out the solution of 4’s to create problems that will equal to 1 – 12.
1 = 44 / 44 or (4+4) / (4+4)
2= 4/4 + 4/4
3 = (4+4+4) /4
4 = 4 x (4 – 4) +4
5 = (4 x 4 + 4)/ 4
6 = 4 x .4 + 4.4
7 = 44 / 4 – 4
8 = 4 + 4.4 - .4
9 = 4/4 +4 +4
10 = 44 / 4.4
11= 4 / .4 + 4/4
12 = (44 + 4) / 4
help and ill give u brainly And points ASAp
Answers
What is the first step when solving the equation 3(3x-5x)+2=-8
Answers
Answer:
Collect the like terms
Step-by-step explanation:
(3x-5x)
3x (-2x) + 2 = -8
On Monday, Mr. Sanchez started reading a 225-page biography. He plans to read 15 pages each day until he finishes the book. Write an explicit function to represent the number of pages he has left to read depending on the day number; Monday is day number 1, Tuesday is day number 2, and so on. Find f(5) and interpret its meaning in this situation.
Answers
Answer:
Step-by-step explanation:
answen
Step-by-step explanationIn a recent election Corrine Brown received 13,436 more votes than Bill Randall. If the total number of votes was 119,444 find the number of votes for each candidate.
Answers
Answer:
53151 for Bill Randall
66565 for Corrine Brown
Step-by-step explanation:
X + X+13414 = 119716
2x + 13414 = 119716
2x = 106302
x=53151
119716 - 53151 = 66565
In a set of 25 aluminum castings, four castings are defective (D), and the remaining twenty-one are good (G). A quality control inspector randomly selects three of the twenty-five castings without replacement to test. The sample space for selecting the group to test contains ____________ elementary events.
Answers
Answer:
The sample space for selecting the group to test contains 2,300 elementary events.
Step-by-step explanation:
There are a total of N = 25 aluminum castings.
Of these 25 aluminum castings, n₁ = 4 castings are defective (D) and n₂ = 21 are good (G).
It is provided that a quality control inspector randomly selects three of the twenty-five castings without replacement to test.
In mathematics, the procedure to select k items from n distinct items, without replacement, is known as combinations.
The formula to compute the combinations of k items from n is given by the formula:
[tex]{n\choose k}=\frac{n!}{k!(n-k)!}[/tex]
Compute the number of samples that are possible as follows:
[tex]{25\choose 3}=\frac{25!}{3!\times (25-3)!}[/tex]
[tex]=\frac{25\times 24\times 23\times 22!}{3!\times 22!}\\\\=\frac{25\times 24\times 23}{3\times 2\times 1}\\\\=2300[/tex]
The sample space for selecting the group to test contains 2,300 elementary events.
Which two options describe the standard deviation?
The variance square
The extreme distance from the mean expected from the data points
The square root of the variance.
The average distance from the mean expected of the data.
Answers
Answer:
The square root of the variance.
The average distance from the mean expected of the data.
Step-by-step explanation:
Standard deviation is a measure of dispersal. It determines the average distance from the mean expected of the data.
Variance is also related to the standard deviationIt is the square root of the variance. The dispersion or variation of a set of values away from the mean is the standard deviation.
Standard deviation = [tex]\sqrt{variance}[/tex]
In Mario kart the special blocks are programmed to deliver 3 green shells for every 11 bananas.if in the last game characters received 104 more bananas than green shells, how many green shells were received?
Answers
Answer:
look im putting this here bc i have to answer 50 questionsfor a quest sdo dont report me ;-;
Step-by-step explanation:
The function h(x)=1/x^2+1 is the result of the composition f(g(x)). If g(x) = x^2+1,what is f(x)? A f(x)=1/square root x B f(x)=1/x C f(x)= 1/x+1 D f(x)=1/x^2+1
Answers
Answer:
Option B is correct
Step-by-step explanation:
Given: [tex]h(x)=\frac{1}{x^2+1}[/tex] is the result of the composition [tex]f(g(x))[/tex].
Also, [tex]g(x)=x^2+1[/tex]
To find: [tex]f(x)[/tex]
Solution:
Take [tex]f(x)=\frac{1}{x}[/tex]
Now check whether [tex]h(x)[/tex] is equal to [tex]f(g(x))[/tex] or not.
First find [tex]f(g(x))[/tex]
[tex]f(g(x))=f(x^2+1)=\frac{1}{x^2+1}[/tex]
Also, [tex]h(x)=\frac{1}{x^2+1}[/tex]
Therefore,
[tex]h(x)=f(g(x))[/tex]
So,
Option B is correct
Answer: B. f(x)=1/x
Step-by-step explanation:
Edge
____
Solve √3x+4 −2=3
Answers
Answer:
[tex]\sqrt[/tex]3x+4=5
Step-by-step explanation:
3x+4=25
3x=21
x=7
Determine the solution set of each equation
A.x²+6x-16=0
b.x²-25=0
c.3y²=3y+60
d.y²+9y-36=0
e.8x²=6x-x²
Answers
Answer:
weak in maths sorry I cannot give answer
Does the following road sign have rotational symmetry?
Yes
No
Answers
Answer:
YES
Step-by-step explanation:
ITS JUST YES
