3. For f(x) = 3x² - 6x + 5, what restriction must be applied so that f-¹(x) is also a function?
Answers
For f(x) = 3x² - 6x + 5, the restriction that must be applied so that f-¹(x) is also a function is that the coefficient of x² should be non-zero, i.e., a ≠ 0.
In general, if f(x) is a function, then its inverse function f-¹(x) exists if and only if the function f(x) is one-to-one. In order to determine the one-to-one nature of the given function, we need to check whether it satisfies the horizontal line test, which is a graphical tool to test the one-to-one nature of a function. If a horizontal line intersects the graph of a function at more than one point, then the function is not one-to-one. On the other hand, if a horizontal line intersects the graph of a function at most one point, then the function is one-to-one.
For the given function, we can find its graph as follows: f(x) = 3x² - 6x + 5
Completing the square, we get: f(x) = 3(x - 1)² + 2This is a parabola with vertex at (1, 2) and axis of symmetry x = 1.The graph of the function is shown below: From the graph, we see that any horizontal line intersects the graph of the function at most once. Hence, the function is one-to-one and its inverse function exists. The inverse function can be found by switching x and y and then solving for y as follows: x = 3y² - 6y + 5
Solving for y using the quadratic formula, we get: y = [6 ± sqrt(6² - 4(3)(5 - x))] / 2(3)y = [3 ± sqrt(9 - 12x + 4x²)] / 3y = (1/3) [3 ± sqrt(4x² - 12x + 9)]
Note that the quadratic formula can only be applied if the discriminant is non-negative. Therefore, we must have:4x² - 12x + 9 ≥ 0Solving this inequality, we get:(2x - 3)² ≥ 0
This is true for all values of x, so there is no restriction on x that must be applied so that f-¹(x) is a function. However, we note that if the coefficient of x² were zero, then the function would not be one-to-one, and hence, its inverse would not exist as a function. Therefore, the restriction is that the coefficient of x² should be non-zero, i.e., a ≠ 0.
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Related Questions
Jackson was solving a two-step equation. Which property justifies the work done between steps one and two?
Answers
The work done between steps one and two is justified by applying the Addition Property of Equality.
In step one, we start with the equation 3x - 2 = 10.
To isolate the variable, we can use the Addition Property of Equality, which allows us to add 2 to both sides of the equation.
This yields 3x = 12 (step two).
The Addition Property of Equality states that if the same value is added to both sides of an equation, the equality remains true.
Hence, the property utilized between these steps is the Addition Property of Equality.
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A farmer sells 9.1 kilograms of pears and apples at the farmer's market.
2
5
of this weight is pears, and the rest is apples. How many kilograms of apples did she sell at the farmer's market?
Express your answer in fraction and decimal form.
Solve on paper. Then check your work on Zearn
Answers
Answer:
the kilograms of apples sold is 33/5 and in decimal is 6.6
Which of the possibilities below represents the mood and figure of the following argument: Some dictators are not warmongers. Since no dictators are egomaniacs, and some egomaniacs are warmongers. Group of answer choices
AEO - 1
IEO - 1
EIO - 4
IOE - 2
Answers
The mood of the argument is categorical and the figure is EIO. To determine the mood and figure of the given argument, we'll first identify the premises and conclusion, and then assign the appropriate categorical proposition to each statement.
The argument can be rewritten as:
Premise 1: Some dictators are not warmongers. (Some D are not W)
Premise 2: No dictators are egomaniacs. (No D are E)
Premise 3: Some egomaniacs are warmongers. (Some E are W)
Conclusion: Some dictators are not warmongers. (Some D are not W)
Now, let's assign the categorical propositions to each statement:
Premise 1: I (Some D are not W)
Premise 2: E (No D are E)
Premise 3: I (Some E are W)
Conclusion: O (Some D are not W)
The mood of the argument is IEO.
Next, we'll determine the figure by looking at the placement of the middle term (E) in the premises:
Premise 2: No D are E (subject-middle term)
Premise 3: Some E are W (middle term-predicate)
The figure is 1 because the middle term is the subject of one premise and the predicate of the other premise.
So, the mood and figure of the argument are IEO-1. Therefore, the correct answer is:
IEO - 1
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In the diagram (triangle)PRQ~ (Triangle) DEF
FIND m
Answers
Answer:
Angle R is equal to 35 and DE is equal to 48
Step-by-step explanation:
In similar triangles, like these, the angles will always be the same to its correspondent. Angle E in the other triangle is 35 degrees so R, its corresponding angle, will be the same. So therefore R is 35.
This is not the same case for sides. There will always be a ratio between both triangles. The ratio in this case is 1.2 to 1. So multiply RP (DE's corresponding side), by 1.2. You will get 48. So therefore DE is equal to 48.
Hope this helped! Please mark me as brainliest!
CAN SOMEONE PLEASE HELP SOLVE THIS TRIANGLE PROOFS SHEET FOR ME
Answers
Answer:
1)
1. Given
2. Vertical Angles Theorem
3. SAS Congruence Postulate
2)
1. Given
2. Definition of Angle Bisector
3. Definition of Perpendicular Lines
4. All right angles are congruent.
5. Reflexive Property
6. ASA Congruence Postulate
3)
1. AC is the perpendicular bisector of PT.
2. Definition of Bisector
3. AC = AC
4. Definition of Perpendicular Bisector
5. <PCA = <TCA
6. SAS Congruence Postulate.
7. PA = TA; CPCTC
Step-by-step explanation:
None necessary.
what is the volume of thtis cube 1 cm 1 cm 1 cm
Answers
Answer:
V= a3
Step-by-step explanation:
i looked it up on my web browser
Hannah has 126 coins. Some quarters and some are dimes. The total value of the coins is $16.50. How many of each coin does Hannah have
Answers
Answer:
Step-by-step explanation:
126 coins
Quarters(25) - Dimes(10)
Has $16.50
Q+D=126
Q*0.25+d*0.10=126
Srry for not answering since the rules say not to give answers away just to help understand more but if you need more help feel free to ask.
If you mix red and white paint in different ratios, you will get different shades of pink paint. If the ratios are equivalent, the shades of pink will be the same. ◦ Mai mixed a batch of pink paint using 5 cups of red paint and 3 cups of white paint. ◦ Priya mixed another batch of pink paint using 7 cups of red paint and 4 cups of white paint. Are these two batches the same shade of pink? Explain.
Answers
Answer:
No,
Step-by-step explanation:
be a light pink and one would be a darker shade of pink.because
The given two batches doesn't have the same shade of pink because the proportion of 7 cups of red : 4 cups of white is greater than 5 cups of red : 3 cups of white.
What is proportion?A proportion is an equation in which two ratios are set equal to each other. In simple words, equivalent fraction are called as proportions.
Now it is given that,
Mai's pink contains- 5 cups of red paint and 3 cups of white paint
So, Ratio of red : white = 5 : 3
⇒ 5/3 = 1.67
Priya;s pink contains- 7 cups of red paint and 4 cups of white paint
So, Ratio of red : white = 7 : 4
⇒ 7/4 = 1.75
Thus, the proportion of 7 : 4 > 5 : 3
Thus, the given two batches doesn't have the same shade of pink because the proportion of 7 cups of red : 4 cups of white is greater than 5 cups of red : 3 cups of white.
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what is the difference between open and closed ended questions
Answers
Open-ended questions allow for a wide range of responses and encourage the respondent to provide detailed and unrestricted answers. Closed-ended questions, on the other hand, provide a limited set of predetermined response options for the respondent to choose from.
Open-ended questions: Open-ended questions are designed to gather qualitative data and elicit more in-depth responses. They allow respondents to express their thoughts, opinions, and experiences in their own words. These questions do not limit the possible answers and provide the opportunity for the respondent to provide unique and individualized responses.
What do you think about the current situation of the economy, for instance?
Closed-ended questions: Closed-ended questions provide a fixed set of response options from which the respondent must choose. These questions are typically used to gather quantitative data and provide more structured and easily quantifiable answers. Closed-ended questions are useful when specific information or specific response options are required.
For instance, "Do you agree or disagree that the economy is in a good place right now?" (with response options: Agree/Disagree/Neutral)
In conclusion, open-ended questions allow for more diverse and subjective responses, providing richer qualitative data, while closed-ended questions provide limited response options and are more suitable for gathering quantitative data. The choice between open-ended and closed-ended questions depends on the research objectives, the type of data needed, and the level of flexibility desired in the responses.
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What is the distance between -34 and 7
Answers
Answer:
should be 41
Step-by-step explanation:
if you count the jumps on the number line, -34 and 7 have 41 jumps between them.
Answer:
41 units
Step-by-step explanation:
Take the absolute value of each number and add them together to find the distance between them.
34+7=41
x²+y²=121
what is the center and raduis of that
Answers
Answer:
Radius: 11Precalculus Examples
Popular Problems Precalculus Find the Center and Radius x^2+y^2=121
x^2+y^2=121
This is the form of a circle. Use this form to determine the center and radius of the circle.
(x−h)^2+(y−k)^2=r^2
Match the values in this circle to those of the standard form. The variable r represents the radius of the circle, h represents the x-offset from the origin, and k represents the y-offset from origin.
r=11
h=0
k=0
The center of the circle is found at (h,k).
Center: (0,0)
These values represent the important values for graphing and analyzing a circle.
Center: (0,0)
Radius: 11
Step-by-step explanation:
Hope it is helpful......
Helpppppp againnn plzzz
Answers
Answer:
x=8.17
Step-by-step explanation:
uztstskycuvkhifig
Answer:
x = 8.17
Step-by-step explanation:
We will be using pythagoras theorem:
\(a^{2} + b^{2} = c^{2}\)
=> \(4.25^{2} + x^{2} = 9.21^{2}\)
=> \(x^{2}\) = 84.8241 - 18.0625 = 66.7616
=> x = \(\sqrt{66.7616}\) = 8.17077719.. = 8.17 (3 s.f)
Study the following program:
x = 1 while True: if x % 5 = = 0: break print(x) x + = 1 What will be the output of this code?
Answers
The output of the following code snippet will be 1, 2, 3, and 4.
The reason is that the while loop runs infinitely until the break statement is executed. The break statement terminates the loop if the value of x is divisible by 5. However, the value of x is incremented at each iteration of the loop before checking the condition. Here is the step-by-step explanation of how this code works:
Step 1: Assign 1 to x.x = 1
Step 2: Start an infinite loop using the while True statement.
Step 3: Check if the value of x is divisible by 5 using the if x % 5 == 0 statement.
Step 4: If the value of x is divisible by 5, terminate the loop using the break statement.
Step 5: Print the value of x to the console using the print(x) statement.
Step 6: Increment the value of x by 1 using the x += 1 statement.
Step 7: Repeat steps 3-6 until the break statement is executed.
Therefore, the output of the code will be: 1 2 3 4.
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Find the line parallel to y = -3x + 1 that passes through (9, 0). Plls Hurry I'm being timed
Answers
Answer:
1,4
Step-by-step explanation:
The temperature reading that is halfway between −10 degrees and 40 degrees is ___ degrees.
If the temperature was −3 degrees in the morning and 8 degrees in the afternoon, the change in temperature during that time is ____ degrees.
If the temperature has been changing at a rate of −1.5 degrees each hour, and it is currently 5 degrees, the temperature 4 hours ago was ___ degrees.
If the temperature continues to change at this rate of −1.5 per hour, and it is currently 5 degrees, the temperature in 4 hours will be ____ degrees.
Answers
Answer:
30, +11, 11, -1
Step-by-step explanation:
-10 + 30 = 10, 40 - 30= 10
-3 + 11 = 8
5 + (1.5 x 4) = 11
5 - (1.5 x 4) = -1
Solve this equation (ANSWER QUICK)
Answers
Answer:
x=8
Step-by-step explanation:
First off, make the equation easier to solve by putting the non-variable numbers to the other side:
-1+0.5x+9.7=12.7
0.5x+9.7-1=12.7
0.5x+8.7=12.7
-8.7 -8.7
0.5x+0=4.0
Then, multiply 0.5x by 2 and 4.0 by 2 getting you the answer of:
1x=8
x=8
Which of the following is NOT true about unit rates?
1. A unit rate relates a quantity to
2. Unit rates can be used to compare two rates.
3. Any rate is equivalent to its unit rate.
4. A unit rate always relates quantities with the same units.
1,2,3,4 is the correct option
Answers
Answer:
4
Step-by-step explanation:
For different problems or situations, the unit might be different (dollars, miles, crocodiles, etc). However, a unit rate can be used to compare two rates and is always equal to any other rate you can write for the same problem. (since 4 is clearly false, I assume 1 is correct, even though it seems to have been cut off).
come up with a situation that you believe could follow a binomial distribution. explain why you believe it is reasonable to expect this situation to fit a binomial distribution.
Answers
A situation that could follow a binomial distribution is the number of heads that occur when flipping a "fair-coin" multiple times.
A "Binomial-Distribution" is defined as a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has the same probability of success.
In this case, flipping a "fair-coin" is an independent trial with two possible outcomes (heads or tails), and the probability of getting a heads on any given flip is 0.5.
Each "coin-flip" is independent of the others, and the probability of getting a heads on any given flip is always the same which is "0.5",
So, this situation fits the requirements for a binomial distribution.
The "binomial-distribution" can then be used to calculate the probability of getting a certain number of heads out of a fixed number of flips. For example, if we flip a coin 10 times, the probability of getting exactly-5 heads can be calculated using the binomial distribution formula.
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at what point does the curve have maximum curvature? y = 8ex
Answers
The curve y = 8e^x has maximum curvature at the point (0, 8).there is no point where the second derivative equals zero,
To determine the point of maximum curvature on the curve y = 8e^x, we need to find the second derivative of the function and identify the point where it equals zero. The second derivative represents the rate of change of the slope, or the curvature, of the curve.
First, we find the first derivative of y = 8e^x with respect to x:
dy/dx = 8e^x
Then, we differentiate again to find the second derivative:
d²y/dx² = d(8e^x)/dx = 8e^x
Setting the second derivative equal to zero:
8e^x = 0
Exponential functions, such as e^x, are never equal to zero for any value of x. Therefore, there is no point where the second derivative equals zero, indicating that the curve y = 8e^x does not have any inflection points or points of maximum curvature.
However, we can determine the point where the curvature is the greatest by considering the graph of y = 8e^x. Since the exponential function e^x is always positive, the curve y = 8e^x is always concave up. This means that the curvature is positive throughout the curve, but there is no specific point of maximum curvature.
In conclusion, the curve y = 8e^x does not have a point of maximum curvature but is always positively curved.
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I have 3 slopes, I need the Y intercept and the slope please. Will give brainliest. You can label them as "picture 1" or "slope 1 = (?)"
Answers
Answer:
Slpe 1
Step-by-step explanation:
Evaluate the following expression. In e^e
A. 2e
B. e
C. 0
D. 1
Answers
Answer:
it's B e......... I think
what is negative 81 plus 97
Answers
Answer:
16
Step-by-step explanation:
Equations with parentheses Solve for b 3-2(b-2)=2-7b3−2(b−2)=2−7b3, minus, 2, left parenthesis, b, minus, 2, right parenthesis, equals, 2, minus, 7, b b =b=b, equals Stuck? Watch a video or use a hint. Report
Answers
Answer:
Can you retype the question it is difficult to understand when copy and pasted
Step-by-step explanation:
Answer:
sorry
Step-by-step explanation:
The correct value of x is 3
Answers
4y+2x=100
Subtract the first equation from the second
3y=105
And 7/8 hours Greg reads 2/3 chapters what’s the unit rate in chapters per hour?
Answers
The unit rate in chapters per hour is 21/16 hours
How to calculate the unit rate?Greg read 7/8 hours in 2/3 chapter
The unit rate can be calculated as follows
7/8= 2/3
1= x
cross multiply both sides
2/3x= 7/8
x= 7/8 ÷ 2/3
x= 7/8 × 3/2
x= 21/16
Hence 21/16 chapters is read in one hour
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Under what operations is the system of polynomials not closed?.
Answers
Answer:
Polynomials form a system similar to the system of integers, in that polynomials are closed under the operations of addition, subtraction, and multiplication.
...
Polynomials: YES or NO
3x½ NO - exponent is not a whole number
2x-3 NO - exponent is not a whole number. Negative exponent places x3 in denominator.
Step-by-step explanation:
Which expression is NOT equivalent to 24?
2x2x2x3
2x2x6
2x2x3x3
2x4x3
Answers
Answer:
2x2x3x3=36...so it is not equivalent to 24
Explanation: When you do the math for all of the other ones it equals 24.
1. 2x2x2x3.
Two times two is four. Then four times two is eight. Then eight times three is 24.
2. 2x2x6
Two times two is four. Then Four times six is 24.
3. 2x2x3x3
Two times two is four. Then four times three is twelve. Finally twelve times three is 36.
4. 2x4x3
Two times four is eight. Then eight times three is 24.
As you can see, the correct expression that DOES NOT equal to(or equivalent to) is the expression: 2x2x3x3
Hope this helps!!!
I have 1/6 + 5/8how do I work it out
Answers
1/6 + 5/8
First find the lcm of 6 and 8
l c m of 6 and 8 =24
convert each fraction to its equivalent with 24 as its denominator
That is;
1/6 = 4/24
5/8 = 15/24
Then add
4/24 + 15/24
= 19/24
{Q#2} Given the circle with equation x+ y =17
a). Determine if the points D(-4, -1) and E(1, 4) are on the circle. Show all work
b). Find the equation of the perpendicular bisector of the chord DE
c). Verify that the perpendicular bisector from part b) passes through the center of the circle
Answers
Try the suggested solution shown on the attached picture.
If the length and width of a rectangle
are 8 and 5, the length of a diagonal is
(A) 89
(B) V39
(C) V789
(D) V13
Answers
Answer:
d
um belo dia abençoado