1. Find the domain and range of the function
f(x)=√1-4x².
Answers
[tex]f(x) = \sqrt{1 - 4{ x}^{2} } [/tex]
[tex]1 - 4x {}^{2} \geqslant 0[/tex]
[tex] - \infty \: \: \: \: \: \: \: \: - 0.5 \: \: \: \: \: \: \: \: \: 0.5 \: \: \: \: \: \: \: \: \infty \\ - \: \: \: \: \: \: \: \: \: \: \: \: \: \:0 \: \: \: \: \: + \: \: \: \:0 \: \: \: \: \: - [/tex]
[tex]domain \\ [ \: -0.5 \: , \: 0.5 \: ][/tex]
[tex]g(x) = 1 - 4x {}^{2} \\ g'(x) = - 8x \\ g'(x) = 0 \\ x = 0 \\ g(0) = 1 - 4(0) = 1 \\ maximum \: = 1[/tex]
[tex]range \\ [ \: 0 \: , \: 1 \: ][/tex]
Answer:
[tex]\textsf{Domain}: \quad \left[-\dfrac{1}{2}, \dfrac{1}{2}\right][/tex]
[tex]\textsf{Range}: \quad [0, 1][/tex]
Step-by-step explanation:
Domain: set of all possible input values (x-values)
Range: set of all possible output values (y-values)
Given function:
[tex]f(x)=\sqrt{1-4x^2}[/tex]
As negative numbers don't have real square roots:
[tex]\implies 1-4x^2\geq 0[/tex]
Therefore, to find the domain, solve the inequality.
Subtract 1 from both sides:
[tex]\implies -4x^2\geq -1[/tex]
Divide both sides by -1 (reverse the inequality):
[tex]\implies 4x^2 \leq 1[/tex]
Divide both sides by 4:
[tex]\implies x^2\leq \dfrac{1}{4}[/tex]
[tex]\textsf{For }\:a^n \leq b,\:\:\textsf{if }n\textsf{ is even then }-\sqrt[n]{b} \leq a \leq \sqrt[n]{b}:[/tex]
[tex]\implies -\sqrt[2]{\dfrac{1}{4}} \leq x \leq \sqrt[2]{\dfrac{1}{4}}[/tex]
[tex]\implies -\sqrt{\dfrac{1}{4}} \leq x \leq \sqrt{\dfrac{1}{4}}[/tex]
[tex]\implies -\dfrac{1}{2} \leq x \leq \dfrac{1}{2}[/tex]
Therefore:
[tex]\textsf{Domain}: \quad \left[-\dfrac{1}{2}, \dfrac{1}{2}\right][/tex]
To find the range, input the endpoints of the domain into the function:
[tex]\implies f\left(-\dfrac{1}{2}\right)=\sqrt{1-4\left(-\dfrac{1}{2}\right)^2}=0[/tex]
[tex]\implies f\left(\dfrac{1}{2}\right)=\sqrt{1-4\left(\dfrac{1}{2}\right)^2}=0[/tex]
To find the limit of the range, find the extreme point(s) of the function by differentiating the function and setting it to zero.
[tex]\implies f(x)=(1-4x^2)^{\frac{1}{2}}[/tex]
[tex]\implies f'(x)=\dfrac{1}{2}(1-4x^2)^{-\frac{1}{2}} \cdot -8x[/tex]
[tex]\implies f'(x)=-\dfrac{4x}{\sqrt{1-4x^2}}[/tex]
Setting it to zero and solving for x:
[tex]\implies -\dfrac{4x}{\sqrt{1-4x^2}}=0[/tex]
[tex]\implies -4x=0[/tex]
[tex]\implies x=0[/tex]
Substitute x = 0 into the function:
[tex]\implies f(0)=\sqrt{1-4(0)^2}=1[/tex]
Therefore, the range is [0, 1]
Related Questions
WHICH SERIES OF TRANSFORMATIONS CAN BE USED TO SHOW THAT ABC~DEF?
Answers
Answer:
A) is correct
Step-by-step explanation:
Find the value needed for the “?” to make the equation true.
d^?/d^11 = d^7
Answer: ? =__
Answers
Answer:
? = 18
Step-by-step explanation:
Answer:
? = 18
Step-by-step explanation:
d^?/d^11 = d^7
d^? = d^11*d^7
d^? = d^(11+7)
d^? = d^18
? = 18
cuál es el número entero de 2° bajo Cero
Answers
Answer:
-2 °C.
Step-by-step explanation:
Usa el signo + para representar temperaturas arriba de cero y el signo – para representar temperaturas bajo cero.
Find the Area of the triangle
6.3 yd
3.4 yd
Answers
3.15 is half of 6.3.
3.15 x 3.4 = 10.71 yds squared
Please help answer !:)
( WILL GIVE BRAINLST)
Answers
i dont know that so u answer it
jfdxaMXDCJINCIZSCKZDXVSJOISACXKISZPIAnhCPKISZJCPKNSPKCN KPZSA
Answers
Answer:
Jassjsjdhjajjjjajajajajjjajajajajajqjqjjqjsjsjajajajajjajajaja
EASY MATH QUESTION Urgent pls
Answers
Answer:
I don't know that is hard
Step-by-step explanation:
You said this was an easy math question.
Answer:
I think it's b
Step-by-step explanation:
ah=b1+b2/2
h=ax(b1+b2)/2
I have to turn this in today!!! Please help me ASAP!
Answers
Answer:
D. 9/5
Step-by-step explanation:
27:15= 9:5
18:10= 9:5
Answer:
D) [tex]\frac{9}{5}[/tex]
Step-by-step explanation:
The scale factor between the shapes is 1.8 or 9/5 as a fraction.
Hope it helps!
Plz guys help me plz
Answers
Answer:
The answer is A!
Step-by-step explanation:
Its correct, I took the quiz :)
Which of the following is most likely the next step in the series ?
Answers
Answer:
A
Step-by-step explanation:
The colour keeps switching from red to blue to red, and red was last in the series. Therefore, a blue pattern should be next in the series.
Each time, a new hand is added to the pattern. Therefore, we add a hand.
Two less than three times a number
Answers
Answer:
3x-2
Step-by-step explanation:
3 times a number is 3x, if we use x as the number
Two less than (3x) is 3x - 2
Me pueden ayudar con la F porfa :)
Answers
Answer: -66
Step-by-step explanation:
[tex](2)-(-3)(-20)-(-8)[/tex]
multiply first {*}
[tex](2)-(60)-(-8)[/tex]
then open parenthesis {()}
[tex]2-60+8[/tex]
then add/subtract {+/-}
[tex]2-60+8=\\2-68=\\-66[/tex]
Sorry that it isn't in Spanish ;-;
Answer:
f. = -50
Step:
(2) - (-3) (-20) - (-8)
2-(-3) (-20)= -58
-58+8
= -50
Divide the complex numbers and express answer in the form (a+bi)
3-4i/8+2i
Answers
Answer:
[tex]\frac{4}{17}-\frac{19}{34}i[/tex]
Step-by-step explanation:
1. Approach
To divide complex numbers, first one must set the problem up as a fraction. The one will multiply the fraction by the complex conjugate. A complex conjugate is a complex number with the imaginary part multiplied by negative one. Since a number over itself equals one, this will ensure that the expression remains true. When one multiplies by the complex conjugate, the difference of squares property complex into play, therefore the denominator simplifies down to a real number (remember one can simplify the powers of (i)). Now all one has to do is divide each element in the complex number in the numerator by the real number in the denominator. If it does not divide evenly, one can just simply the fraction.
2. Solving
Set up as a fraction
[tex]\frac{3-4i}{8+2i}[/tex]
Multiply by the complex conjugate
[tex]\frac{3-4i}{8+2i}*\frac{8-2i}{8-2i}[/tex]
Simplify
[tex]\frac{(3-4i)(8-2i)}{(8+2i)(8-2i)}[/tex]
[tex]\frac{24-6i-32i+8i^{2}}{64-4i^{2}}[/tex]
[tex]\frac{24-6i-32i-8}{64-4(-1)}[/tex]
[tex]\frac{16-38i}{64+4}[/tex]
[tex]\frac{16-38i}{68}[/tex]
[tex]\frac{8-19i}{34}[/tex]
It takes Steven 12.25 minutes to run 1.4 miles. At this rate. how long will it take him to run 2.6 miles?
Answers
What I did was
12.25 divided by 1.4 =8.75
Then
8.75*2.6=22.75
The speed is the distance covered by an object at a particular time. The time it will take for Steven to cover a distance of 2.6 miles is 22.75 minutes.
What is speed?The speed is the distance covered by an object at a particular time. Therefore, it is the ratio of distance and time.
Speed = Distance / Time
Given that it takes Steven 12.25 minutes to run 1.4 miles. Therefore, the rate or speed at which Steven runs is,
Speed = Distance / Time
Speed of Steve = 1.4 miles / 12.25 minutes
Now, if at the same rate Steven runs, then the time it will take for Steven to run a distance of 2.6 miles is,
Speed = Distance / Time
Time = Distance / Speed
Time = 2.6 miles / (1.4 miles / 12.25 minutes)
Time = (2.6 miles × 12.25 minutes)/1.4 miles
Time = 22.75 minutes
Hence, the time it will take for Steven to cover a distance of 2.6 miles is 22.75 minutes.
Learn more about Speed here:
https://brainly.com/question/15100898
#SPJ2
A rectangular swimming pool measures 40 ft by 60 ft and is surrounded by a path of uniform width around the four edges. The perimeter of the rectangle formed by the pool and the surrounding path is 248 ft. Determine the width of the path.
Answers
Answer:
6ft
Step-by-step explanation:
Lets x = the width of the path the surrounding path will add 2x to the pool dimension,therefore over all dimesion: (2x+40) by (2x+60) the overall perimeter (2x + 40) + 2(2x+60) = 248 Simplify divide by 2, result (2x+40) +(2x+60) = 124
Combine like terms 2x + 2x +40 +60 =124
4x + 100 = 124
4x=124 = 100
4x=24
x=24/4
x= 6ft is the width of the path
check this by finding the perimeter with these values; 2x = 12 ft
2 ( 12 + 40 ) + 2( 12 +60 )
2( 52 + 2(72)
104 + 144 = 248; confirms our solution of x= 6 ft
( NEEDED ASAP! )
Capon City has 5 more than 3 times as many residents as Harbor City.
Create an algebraic expression to represent the number of residents in Capon City. Use r in your expression where r represents the number of residents in Harbor City
Answers
Answer:
y=3r+5
Step-by-step explanation:
Break up the sentance.
Capon city has 5 more than 3 times as many residents as harbor city.
The sentance above can be written like this: y=5+3x. Since they want r to represent the number of people in Harbor city, use r in place of x.
Capon city=y
has=equal sign
5 more=+5
3 times as many=3r
y=3r+5
Suppose your friends have the following ice cream preferences: 31% of your friends like chocolate (C). The remaining do not like chocolate. 15% of your friends like sprinkles (S) topping. The remaining do not like sprinkles. 5% of your friends like Chocolate (C) and also like sprinkles (S). Of the friends who like sprinkles, what proportion of this group likes chocolate
Answers
Answer:
0.3333 = 33.33% of this group likes chocolate
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.
Of the friends who like sprinkles, what proportion of this group likes chocolate:
So,
Event A: Likes sprinkles
Event B: Likes chocolate
15% of your friends like sprinkles (S) topping.
This means that [tex]P(A) = 0.15[/tex]
5% of your friends like Chocolate (C) and also like sprinkles (S).
This means that [tex]P(A \cap B) = 0.05[/tex]
So
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.05}{0.15} = 0.3333[/tex]
0.3333 = 33.33% of this group likes chocolate
Simplify -12+8+5+(-9)
Answers
−12+8+5-9
Simplify -12+8 to -4
-4 + 5 — 9
Simplify
−4 + 5 −4+5 to 1
1 — 9
Simplify
-8 is your answer
Answer:
The answers is -8
Step-by-step explanation:
Remove parentheses.
-12 + 8 + 5 - 9
Add -12 and 8
-4 + 5 -9
Add -4 and 5
1 - 9
Subtract 9 from 1
-8
The answers is -8
Hope this helps
Encuentre el resultado de la suma de las fracciones, simplifique la fraccion. 6/7+2/8=
Answers
Answer:31/28
Step-by-step explanation:
Answer:
1 3/28
Step-by-step explanation:
Express using algebra A increased by 50% and then decreased by 50% of that
Answers
Answer:
.5(1.5A)
Step-by-step explanation:
Increasing by 50% is the same as multiplying by 1.5. Decreasing by 50% is the same as multiplying by .5. Multiply by 1.5 first then surround with parentheses and multiply by .5.
Answer:
.5(1.5A)
Step-by-step explanation:
Increasing by 50% is the same as multiplying by 1.5. Decreasing by 50% is the same as multiplying by .5. Multiply by 1.5 first then surround with parentheses and multiply by .5.
What is the sum of the polynomials? (6x+7x+x^2)+(2x^2-3x)
Answers
Answer:
3x^2 + 10x
Step-by-step explanation:
1) Dan ate 1/12 of a pumpkin, while Sandy ate 1/2 of a pumpkin. In total, how much pumpkin did these two eat?
Answers
Answer:
1 pumpkin
Step-by-step explanation:
1/2 + 1/2 is 1. They ate 1 pumpkin in total.
1/2+1/2( half + half)=1
What is the volume?
7 mm
7 mm
1 mm
Answers
Answer:
(7)(7)(1)=49 mm^3
Step-by-step explanation:
Answer:
49mm³
Step-by-step explanation:
[tex]v = 7mm \times 7mm \times 1mm = 49m {m}^{3} [/tex]
Write a second inequality with the same meaning -12<= a
Answers
Answer:
a[tex]\geq[/tex]-12 or -48[tex]\leq[/tex]4a
Step-by-step explanation:
Beth’s brother is 140% of Beth’s height. If Beth’s brother measures 140cm, how tall is Beth?
Answers
Answer:
100 cm
Step-by-step explanation:
let x be the height of Beth
and y be the height of Beth's Brother
By 1st Condition
y = 140% * x
y = (140/100) * x
y = 1.4 x ..........................(Eq 1)
By 2nd Condition
y = 140
Put the value of y in the equation 1
140 = 1.4 *x
x = 140 / 1.4
x = 100 cm
Is collecting Taxes a
Delegated power
Reserved power
Concurrent power
Answers
Answer: C.concurrent power
Step-by-step explanation: Such powers are called concurrent powers. These include the power to tax, spend, and borrow money. State governments operate their own judicial systems, charter corporations, provide public education, and regulate property rights.
Which graph represents the solution to the compound inequality?
–6 < 3x – 12 ≤ 9
A number line with an open circle at 2 with a bold line pointing to the left. A point at 7 with a bold line pointing to the right.
A number line with an open circle at 2 with a bold line pointing to the right ending at the point at 7.
A number line with an open circle at negative 6 with a bold line pointing to the left. A point at 9 with a bold line pointing to the right.
A number line with an open circle at negative 6 with a bold line pointing to the right ending at the point at 9.
Answers
Answer:
B) The second one. A number line with an open circle at 2 with a bold line pointing to the right ending at the point at 7.
Step-by-step explanation:
B) The second one. A number line with an open circle at 2 with a bold line pointing to the right ending at the point at 7.
Answer:
B
Step-by-step explanation:
edge 2022
1. A square is altered so that one dimension is
increased by 7 meters and the other dimension
is increased by 4 meters. If the area of the
resulting rectangle is 108 square meters, find the
area of the original square.
Answers
11z > -33 help please
Answers
11z / 11 > 33 / 11
z > 3
Hope it helps!
Please help this is due today!
Calculate the unknown missing angle measures.
Do not answer this with one answer ,blank, or a ridiculous answer.... this is serious please.
Answers
Answer:
For ABC A= (-3,3) B= (0.2) and C= (-3,1)
For DFE D= (0,6) F= (2,6) and E= (2,3)
Step-by-step explanation:
I believe tha name of a sequence of transformation will be translation and the Reflection
Hope this Helped