Answer:
The inverse of f(x) is [tex]f^{-1}[/tex](x) = ± [tex]\sqrt{\frac{x+\frac{11}{4}}{3}}[/tex] + [tex]\frac{1}{2}[/tex]
Step-by-step explanation:
To find the inverse of the quadratic function f(x) = ax² + bx + c, you should put it in the vertex form f(x) = a(x - h)² + k, where
h = [tex]\frac{-b}{2a}[/tex] k is the vlue f at x = h∵ f(x) = 3x² - 3x - 2
→ Compare it with the 1st form above to find a and b
∴ a = 3 and b = -3
→ Use the rule of h to find it
∵ h = [tex]\frac{-(-3)}{2(3)}[/tex] = [tex]\frac{3}{6}[/tex] = [tex]\frac{1}{2}[/tex]
∴ h = [tex]\frac{1}{2}[/tex]
→ Substitute x by the value of h in f to find k
∵ k = 3( [tex]\frac{1}{2}[/tex])² - 3( [tex]\frac{1}{2}[/tex]) - 2
∴ k = [tex]-\frac{11}{4}[/tex]
→ Substitute the values of a, h, and k in the vertex form above
∵ f(x) = 3(x - [tex]\frac{1}{2}[/tex])² + [tex]-\frac{11}{4}[/tex]
∴ f(x) = 3(x - [tex]\frac{1}{2}[/tex])² - [tex]\frac{11}{4}[/tex]
Now let us find the inverse of f(x)
∵ f(x) = y
∴ y = 3(x - [tex]\frac{1}{2}[/tex])² - [tex]\frac{11}{4}[/tex]
→ Switch x and y
∵ x = 3(y - [tex]\frac{1}{2}[/tex])² - [tex]\frac{11}{4}[/tex]
→ Add [tex]\frac{11}{4}[/tex] to both sides
∴ x + [tex]\frac{11}{4}[/tex] = 3(y - [tex]\frac{1}{2}[/tex])²
→ Divide both sides by 3
∵ [tex]\frac{x+\frac{11}{4}}{3}[/tex] = (y - [tex]\frac{1}{2}[/tex])²
→ Take √ for both sides
∴ ± [tex]\sqrt{\frac{x+\frac{11}{4}}{3}}[/tex] = y - [tex]\frac{1}{2}[/tex]
→ Add [tex]\frac{1}{2}[/tex] to both sides
∴ ± [tex]\sqrt{\frac{x+\frac{11}{4}}{3}}[/tex] + [tex]\frac{1}{2}[/tex] = y
→ Replace y by [tex]f^{-1}[/tex](x)
∴ [tex]f^{-1}[/tex](x) = ± [tex]\sqrt{\frac{x+\frac{11}{4}}{3}}[/tex] + [tex]\frac{1}{2}[/tex]
∴ The inverse of f(x) is [tex]f^{-1}[/tex](x) = ± [tex]\sqrt{\frac{x+\frac{11}{4}}{3}}[/tex] + [tex]\frac{1}{2}[/tex]
Plz help me ASAP, Plz show the working so i know you did it, Thank you so much
Step-by-step explanation:
Let's use PEMDAS here
P: N/A
E: To transform [tex]5^{x}[/tex] to [tex]5^{3x}[/tex], we put [tex]5^{x}[/tex] to the power of 3.
M: We multiply 2 to [tex]5^{3x}[/tex], stretching the function by 2
M: We multiply -1 to [tex]2*5^{3x}[/tex], flipping the function vertically over x=0 (this is interchangeable with multiplying 2)
D: N/A
A: We add 8, moving the function to the right by 8
S: N/A
If f(x) = 2x − 1 and g(x) = f(x + 1), how does the graph of g compare with the graph of f?
Cancer is caused by abnormal growth is due to a genetic mutation. Which of the following can cause this genetic mutation? (Choose all that apply)
a) caused by exposure to radiation
b) can be inherited
c) acquired with age after many rounds of cell division
d)acquired from contact with a person who has cancer
Two hiking groups made the purchases shown below. What is the cost of each item. GROUP 1: 6 muffins, 12 granola bars, Total=$26.70 GROUP 2: 8 muffins, 15 granola bars, Total=34.00
Answer:
Price of each muffin = $1.25
Price of each granola bar = $1.6
Step-by-step explanation:
Group 1:
6 muffins, 12 granola bars
Total money paid = $26.70
Group 2:
8 muffins, 15 granola bars,
Total money paid = $34.00
To find:
Price of each item?
Solution:
Let the price of one muffin = $[tex]x[/tex]
Let the price of one granola bar = $[tex]y[/tex]
As per group 1, the equation can be written as:
[tex]6x+12y=26.70\\\Rightarrow x+2y=4.45 .... (1)[/tex]
As per group 2, the equation can be written as:
[tex]8x+15y = 34[/tex] .... (2)
Let us use elimination method to solve for [tex]x[/tex] and [tex]y[/tex].
Multiplying equation (1) with 8 and subtracting (2) from it:
[tex]16y-15y = 35.6 - 34\\\Rightarrow y = 1.6[/tex]
Using equation (1):
[tex]x+3.2=4.45\\\Rightarrow x = 1.25[/tex]
Therefore, the answer is:
Price of each muffin = $1.6
Price of each granola bar = $1.25
Calculate The Constant of Proportionality (k) and Write an Equation for the line in the Graph Below:
Answer:
k = ½
Equation => [tex] y = \frac{1}{2}x [/tex]
Step-by-step explanation:
Constant of proportionality (k) = y/x
Using a point on the graph, (6, 3),
k = y/x = 3/6 = ½
k = ½
Equation of the line:
Plug in the value of k into [tex] y = kx [/tex] (slope-intercept form of equation for proportional relationship)
Thus:
[tex] y = \frac{1}{2}x [/tex]
Select the correct answer. Liam chooses an investment plan that requires him to make an initial investment of $8,000. Then he contributes equal monthly investments of $300 for the next 10 years. What function represents the amount in dollars that Liam has invested in terms of the number of months since he made the initial investment? A. Y = 8,000 − 300x B. Y = -8,000 + 300x C. Y = 8,000 + 300x D. Y = 300 − 8,000x E. Y = 300 + 8,000x
Answer: C. Y = 8,000 + 300x
Step-by-step explanation:
Given: Initial investment = $8,000
Monthly investment = $300
Let x = the number of months since he made the initial investment.
Y= the amount in dollars that Liam has invested.
Now , Amount invested = Initial investment + (Monthly investment) x ( number of months)
Y = 8000 + 300x
Hence, the required function : Y = 8000 + 300x
So, the correct option is C. Y = 8,000 + 300x
Answer:
The answer is c
Step-by-step explanation:
yuh
Write an equation of the line that passes through (-1,3) and is parallel to the line y = 2x + 2.
y =
to solve the following equation , 7x - 2 = 12 , which of the choices below is a correct method to find the value of X
The Answer:the answer is D
Step-by-step explanation:
Can someone help me with this question? 12-x=5
Answer:
x=7
Step-by-step explanation:
12-x=5
Subtract 12 from both sides.
−x=−7
Divide both sides by -1
x=7
Heather is putting money into a savings account. She starts with 450 in the savings account, and each week she adds 60.
Let S represent the total amount of money in the savings account (in dollars), and let W represent the number of weeks Heather has been adding money. Write an equation relating S to W. Then use this equation to find the total amount of money in the savings account after 13 weeks.
Answer:
1230
Step-by-step explanation:
60 times 13 you get the answer of 780 then you add 450 to it and boom
A stand at a farmer's market sells different types of apples, as shown in the table. Which type of apple costs the least per pound?
Answer:
DATTEBAYOOO
It's just a pic of Naruto Uzumaki, BELEIVE IT
Step-by-step explanation:
IF YOU GOOD AT MATH PLZ HELP !!! Algebra II
Answer:???
Can somebody tell me what is 30% of 240
Answer:
72
Step-by-step explanation:
Answer:
first write 30% as fraction
30/100 then "of" means multiplication
30/100×240
=72
can i have brainliest if it helped
Step-by-step explanation:
which expression is the sum of two terms
Answer:
are there choices or a picture for this question?
Logarithms and stuff, please help
Answer:
[tex]\displaystyle t=-25\log\left(\frac{20}{30}\right)[/tex]
Step-by-step explanation:
Logarithms
The model that predicts the percentage of voters (V) as a function of the number of years (t) is:
[tex]V(t)=100-30\cdot e^{-0.04t}[/tex]
It's required to find in how many years the value of V will be V=80%. The equation to solve is:
[tex]100-30\cdot e^{-0.04t}=80[/tex]
Subtracting 100:
[tex]-30\cdot e^{-0.04t}=80-100=-20[/tex]
Dividing by -30:
[tex]\displaystyle e^{-0.04t}=\frac{20}{30}[/tex]
Applying logs:
[tex]\displaystyle \log( e^{-0.04t})=\log\left(\frac{20}{30}\right)[/tex]
For logs property:
[tex]\displaystyle -0.04t\log(e)=\log\left(\frac{20}{30}\right)[/tex]
log(e)=1. Solving for t:
[tex]\displaystyle t=\frac{\log\left(\frac{20}{30}\right)}{-0.04}[/tex]
Since 0.04=1/25:
[tex]\boxed{\displaystyle t=-25\log\left(\frac{20}{30}\right)}[/tex]
Allison is evaluating the expression -5-(-2) using integer tiles. She begins with 5 negative tiles as shown
VX
What should Allison do next to find the value of the expression?
remove 2 negative tiles
remove 3 negative tiles
add 2 negative tiles
Add 7 negative tiles
Answer:
its B. -3
Step-by-step explanation:
Remove 3 negative tiles .
Can I get brainliest??????
Answer:
Step-by-step explanation:
its b
Eighty students try out for the school play. Of these students, 60% are given roles in the play. Males are assigned 25% of the roles in the play. How many males have a role in the play?
Answer:
20
Step-by-step explanation:
(1/4) × 80
=20
there fore, 20 males have role in the pkay
Answer:
Solution in photo :)
Step-by-step explanation:
I need to find the area. Please help thanks:)
Answer:
area = 8.037 m²
solve for x and graph
Answer:
x≥5
Step-by-step explanation:
A biologist investigates the growth of a special flower. It is estimated that
the height (in cm) of the flower is given by f(t) = at² + bt + C, where t is
the number of days passed since the start of the investigation. It is given
that the height of the flower first increases and then decreases.
What is the domain of f(t)?
Answer:
The domain of this function is:
Domain f(t): t ≥ 0
or
Domain f(t): {x ∈ R | t ≥ 0}
Step-by-step explanation:
Given the function of the height (in cm) of the flower
f(t) = at² + bt + C
Here,
t is the number of days passed since the start of the investigation.f(t) represents the height (in cm)Given that height of the flower first increases and then decreases.
The current function represents the quadratic function because it is of the form. Hence, the graph would be a curve.
f(x) = ax²+ bx + c
where a, b and c are the real numbers and a≠0
Normally the domain of quadratic is the set of all real numbers, but here in this case x-values consist of the time in terms of the number of days.
As the time in terms of a number of days can not be negative. Therefore, we conclude that:
The domain of this function is:
Domain f(t): t ≥ 0
or
Domain f(t): {x ∈ R | t ≥ 0}
Help plzzzzzz it says 5/2 is wrong and I’ve tried 2/5 too both are wrong
Answer:
2.5 or 2 1/2
Step-by-step explanation:
What is the difference of 3.14 and the sum of 5.99 and 0.65?
Answer:
3.5
Step-by-step explanation:
5.99+0.65=6.64
6.64-3.14=3.5
Explain the difference between (-5) to the power 2 and -5 to the power 2 .
Answer:
(-5) to the power of 2 is 25 and -5 to the power to 2 is -25 because minus in not multiplying by antohrer minus
Step-by-step explanation:
A circle has a center at the origin and the point (3,4) is on the circle
Answer: 5 units
Step-by-step explanation:
The standard form of equation of circle:
[tex](x-h)^2+(y-k)^2=r^2[/tex] , where (h,k) = center of circle, r= radius.
Given : Center of circle= origin = (0,0) =(h,k)
Point on circle = (3,4)
Put x= 3 , y=4 , h=0 , k=0 , then the equation will become
[tex](3-0)^2+(4-0)^2=r^2\\\\\Rightarrow\ 9+16=r^2\\\\\Rightarrow\ 25=r^2\\\\\Rightarrow\ r=5[/tex]
Hence, the radius is 5 units long.
What is the scale factor of the dilation?
Answer:
3
Step-by-step explanation:
the distance from the point of dilation to point C is 3
the distance from the point of dilation to point A is 9
9/3 = 3
hope this helps, pls mark brainliest :D
Plz answer answer this I’ll give brainliest
Step-by-step explanation:
6 ◇ 3
= (6² - 3) / 3
= 33 / 3
= 11.
Question 18 will give brainliest
Answer:
as below
Step-by-step explanation:
Please help due in 10 minutes
Answer:
The 2nd one should be the correct answer
answer: the 2nd option. (aka: 2(3x+5)
if : 3x+3x+5+5= 6x+10
THEN: 2(3x+5)
HOW: 2 times 3x.
2 times 5
6x+10.
hope i helped! we just passed this subject in my class. Good luck!!
Drag each label to the correct answer
Answer:
um
Step-by-step explanation:
if the equation of L1 is 4x-2y=6 what is the x intersept?
Write the equation of L in slope intersept form
Write the slope of L.
Answer:
The x intercept: (1.5, 0)
The slope intercept form: y = 2x - 3
The slope: m = 2
Step-by-step explanation:
x intercept means y=0
4x - 2y = 6
4x - 2·0 = 6
4x = 6
x = 1.5
The slope intercept form is: y = mx + b
4x - 2y = 6 {subtract 4x from both sides}
- 2y = - 4x + 6 {dividing both sides by (-2)}
y = 2x - 3 ← slope intercept form
Slope is the number by x, in slope intercept form of line's equation