Answers
Answer:
Domain: [2, ∞)Range: [0, ∞)Continuity: Function is continuous on its domain [2, ∞).Minimum point at (2, 0).Increasing function: (2, ∞)Symmetry: Neither as there is no symmetry about the y-axis and no origin symmetry.Step-by-step explanation:
Given function:
[tex]y=\sqrt{x-2}[/tex]
DomainThe domain is the set of all possible input values (x-values).
As the square root of a negative number cannot be taken, the domain of the given function is restricted.
[tex]x-2\geq 0 \implies x\geq 2[/tex]
Therefore, the domain of the given function is [2, ∞).
RangeThe range is the set of all possible output values (y-values).
As the square root of a negative number cannot be taken, the range of the given function is restricted.
Therefore, the range of the given function is [0, ∞).
ContinuityA function f(x) is continuous when, for every value [tex]c[/tex] in its domain:
[tex]\text{$f(c)$ is de\:\!fined \quad and \quad $\displaystyle \lim_{x \to c} f(x) = f(c)$}[/tex]
Therefore, the function is continuous on its domain [2, ∞).
Maximums and MinimumsStationary points occur when the gradient of a graph is zero.
Therefore, to find the x-coordinate(s) of the stationary points of a function, differentiate the function, set it to zero and solve for x.
[tex]\begin{aligned}y & = \sqrt{x-2}\\& = (x-2)^{\frac{1}{2}\\\implies \dfrac{\text{d}y}{\text{d}x} & = \dfrac{1}{2}(x-2)^{-\frac{1}{2}}\\ & = \dfrac{1}{2\sqrt{x-2}}\\\\\dfrac{\text{d}y}{\text{d}x} & = 0\\\implies \dfrac{1}{2\sqrt{x-2}} &= 0\\\dfrac{1}{2} &\neq 0\end{aligned}[/tex]
Therefore, there are no stationary points.
As the domain is restricted and f(x) ≥ 0, the minimum point of the function is at point (2, 0).
Increasing/Decreasing Function[tex]\textsf{A function is \textbf{increasing} when the \underline{gradient is positive}} \implies \dfrac{\text{d}y}{\text{d}x} > 0[/tex]
[tex]\textsf{A function is \textbf{decreasing} when the \underline{gradient is negative}} \implies \dfrac{\text{d}y}{\text{d}x} < 0[/tex]
Increasing
[tex]\begin{aligned}\dfrac{\text{d}y}{\text{d}x} & > 0 \\ \implies \dfrac{1}{2\sqrt{x-2}} & > 0 \\ \dfrac{1}{\sqrt{x-2}} & > 0\\ \textsf{If $\frac{1}{a} > 0$ then $a > 0$}: \\ \implies \sqrt{x-2} & > 0\\ x-2 & > 0\\ x & > 2\end{aligned}[/tex]
Decreasing
[tex]\begin{aligned}\dfrac{\text{d}y}{\text{d}x} & < 0 \\ \implies \dfrac{1}{2\sqrt{x-2}} & < 0 \\ \dfrac{1}{\sqrt{x-2}} & < 0\\ \textsf{If $\frac{1}{a} < 0$ then $a < 0$}: \\ \implies \sqrt{x-2} & < 0\\ \textsf{As $\sqrt{x}\geq 0$, no solution.}\end{aligned}[/tex]
Therefore:
The function is increasing in the interval (2, ∞).Symmetry[tex]\textsf{A function is \underline{even} when $f(x) = f(-x)$ for all $x$.}[/tex]
[tex]\textsf{A function is \underline{odd} when $-f(x) = f(-x)$ for all $x$.}[/tex]
As there is no symmetry about the y-axis (even symmetry) and no origin symmetry (odd symmetry), the symmetry of the function is neither.
Answer:
Domain: [2, ∞)
Range: [0, ∞)
Continuity: Function is continuous on its domain [2, ∞).
Minimum point at (2, 0).
Increasing function: (2, ∞)
Symmetry: Neither as there is no symmetry about the y-axis and no origin symmetry.
Related Questions
What is the slope of the line that passes through the points (9, 1) and (10,−1)? Write your answer in simplest form.
Answers
Slope of the point is 11/8.
here we use the slope equation so -1-10 would be -11, Then 1-9 is -8 and then -11 divided by -8 is a positive 11/8.
given;
x1=9
x2=1
y1=10
y2=-1
slope=y2-y1/x2-x1
=-1-10/1-9
= 11/8
therefore, the slope of the line that passes through the points ; 11/8.
The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run"
there are 4 types of slope,
negative, positive, zero, and undefined. as x increases
Whenever the equation of a line is written in the form y = mx + b, it is called the slope-intercept form of the equation. The m is the slope of the line. And b is the b in the point that is the y-intercept (0, b)..
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If the measure of A is 28 less than three times the measure of B. A and B are a linear pair. Find the measure of A and B.
Answers
Measure of linear pair A and B are 86 and 38 respectively.
Let the measure of B = x
And measure of A = 3x – 28
As A and B are linear pair. Therefore their sum will be equal to 180
That is numerically,
A + B = 180
Putting the required values of A and B we will get
3x – 28 + x = 180
4x = 180-28
4x = 152
x = 38
Hence the measure of A = 3*38 – 28 = 86
And the measure of B = x = 38
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Marty doesn't have any money saved and he does not want to wait to buy a car . He goes to the dealership and takes out a loan for the $15,000 car . His payments will be $ 500 per month for 3 years (36 months ) . How much will Marty pay for his $ 15,000 car ?
Answers
Marty will have to pay $18,000 for his $15,000 car.
While taking out a loan, the amount taken has to be repaid along with an interest. Thus, the amount paid back by the borrower is always greater than the initial amount that the borrower gets from the lender as a loan.
Here, we are given that Marty has to pay $500 per month for 3 years to repay the loan for his $15,000 car.
Thus, for one month he pays = $500
and also we know that 1 year = 12 months
⇒ 3 years = 12(3) = 36 months
Therefore, for 36 months Marty will have to pay,
500 × 36 = 18,000
Thus, Marty will have to pay $18,000 for his $15,000 car.
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compare the following numbers
2.7×1012;1.5×108
Answers
2.7×1012 (i.e.2,732.4) is greater than 1.5×108 (i.e. 162)
Given two numbers
1)2.7×1012
2) 1.5×108
Two numbers can be equal or one is greater than the other or one is less than the other. there will be no other case rather than these three.
1)2.7×1012=2,732.4
2) 1.5×108=162
The value of the 1st number is 2,732.4 and the value of the 2nd number is 162
1st given number is greater than the 2nd number i.e. 2,732.4>162
Therefore,2.7×1012 (i.e.2,732.4) is greater than 1.5×108 (i.e. 162)
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Solve for x on the triangle
Answers
The value of x in given diagram is x=3;
The entire area filled by a triangle's three sides on a two-dimensional plane is referred to as the triangle's area. The fundamental equation for a triangle's area is A = 1/2 b h, where b and h are the product of the triangle's base and height. All triangle shapes, including scalene, isosceles, and equilateral triangles, may be calculated using this formula. It is important to keep in mind that a triangle's base and height are perpendicular to one another.
Area of triangle SXY = 1/2 × SX × (x-2);
Area of triangle STR = 1/2 × ST × (x+3);
Area of trapezium XTRY = 1/2 × (x+3+x-2) × XT;
Area of triangle SXY = Area of triangle STR - Area of trapezium XTRY;
SX × (x-2) = ST × (x+3) - (2x + 1) × XT;
As; SX = TX = ST/2;
(x-2)/2 = (x+3) - (2x+1)/2;
x - 2 = 2x + 6 -2x -1
x = 3;
A triangle is a closed figure that has three vertices, three sides, and three angles. It is one of the most fundamental geometric forms, and it is represented by the symbol△. Triangles in mathematics can be categorized according to their sides and angles into many categories.
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Find the range please
Answers
Answer:
8
Step-by-step explanation:
The range of a data set in statistics is the difference between the largest and the smallest values. While range does have different meanings within different areas of statistics and mathematics, this is its most basic definition, and is what is used by the provided calculator. Using the same example:
2,10,21,23,23,38,38
38 - 2 = 36
The range in this example is 36. Similar to the mean, range can be significantly affected by extremely large or small values. Using the same example as previously:
2,10,21,23,23,38,38,1027892
The range, in this case, would be 1,027,890 compared to 36 in the previous case. As such, it is important to extensively analyze data sets to ensure that outliers are accounted for.
Answer: Range is all the values of the y-axis therefore, the range of the graph is {0,4,6,8}
the question is in the picture below
Answers
Answer: Answer 2, Triangle A is dilated from the origin by a scale of 1/2 to obtain triangle B
Step-by-step explanation:
A certain hybrid car can travel 1 9/11 times as far as a similar no hybrid car will on one gallon of gasoline . If the non hybrid car can travel 33 miles per gallon of gasoline, how far can the hybrid travel on 4/5 gallon of gasoline ?
Answers
On 4/5 gallon of gasoline, hybrid car will travel a distance of 48 miles.
How to solve Algebra Word Problems?Suppose, non hybrid car travel "x" distance in 1 gallon of gasoline
Then, hybrid will travel " 1 ⁹/₁₁ * x" distance in 1 gallon of gasoline
Given that, non hybrid car travel distance per gallon is; x = 33 miles per gallon
Then, hybrid car travel, 1 ⁹/₁₁ * x = 1 ⁹/₁₁ * x * 33 = 60 miles per gallon
On 1 gallon of gasoline hybrid car travel 60 miles
On 4/5 gallon of gasoline hybrid car will travel = 4/5 * 60 = 48 miles
On 4/5 gallon of gasoline, hybrid car will travel 48 miles of distance.
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On a website selling gifts, the total cost is given by the formula:
Total cost (£) = number of items × 12 + 5
Work out the total cost for 4 items.
Answers
Answer:
4x12 equals 48 therefore 48+5= 53......
Solve |5x+1|<-2 and write the solution in interval notation.
Answers
If you meant to say [tex]|5x+1| < -2[/tex], then there are no solutions.
This is because the |5x+1| is never negative. This applies to the result of any absolute value function. Absolute value represents distance on a number line. Negative distance is not possible.
---------------------------------------------------------------
If you meant to say [tex]|5x+1| \le 2[/tex], then the steps are shown below
[tex]|5x+1| \le 2\\\\-2 \le 5x+1 \le 2\\\\-2-1 \le 5x+1-1 \le 2-1\\\\-3 \le 5x \le 1\\\\-3/5 \le 5x/5 \le 1/5\\\\-3/5 \le x \le 1/5\\\\-0.6 \le x \le 0.2\\\\[/tex]
The interval notation would be [-3/5, 1/5] in fraction form
That is equivalent to [-0.6, 0.2] in decimal form.
Use square brackets to include each endpoint.
Find the domain. Y= 1-7x / 8-|3x-15|
Answers
The domain is written in the form { x : x ≠ 7/3 }
Given data
Y= 1-7x / 8-|3x-15|
How to solve for the domainThe domain refers to the values of x, from the equation we solve for the possible values that will not make the equation undefined.
For the equation to be undefined the denominator should be zero hence we solve for the value of x that will make the equation, to have a denominator equal to zero and the domain will be any other values of x except the value that gives the denominator equal to zero,
The denominator is 8-|3x-15|
solving for the value of x that gives the equation that makes the equation equal to zero
8-|3x-15| = 0
3x - 15 = -8
3x = -8 + 15
3x = 7
x = 7/3
The domain is written in the form { x : x ≠ 7/3 }
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explain how to write 50,000 using exponents
Answers
Explanation:
Place the decimal point between the first two digits to get 5.0
Then we must move the decimal point 4 spots to the right to get back to 50,000
So that's where the "4" comes in with the 10^4
This means 50,000 = 5 * 10^4 which is in scientific notation.
The answer in scientific notation is 5 times 10^4
The exponent of four tells you to multiply 10 by itself 4 times.
10 x 10 = 100 x 10 = 1000 x 10 = 10000
And 10000 times 5 is equal to 50,000.
The variables y and x have a proportional relationship, and y = 12 when x = 5.
What is the value of y when x = 8?
Answers
Y= 19.2
Answer:
y = 19.2
Step-by-step explanation:
We know there is relationship between y and x, so according to the formula [tex]y = kx[/tex]
k = 12/5
we get one function y = 2.4x
when x = 8, input it into the function,
y = 2.4×8 = 19.2
An airplane is flying at a speed of 300 mi/h at an altitude of one mile. The plane passes directly above a radar station at time
t = 0.
Answers
The horizontal distance that the plane has flown can be represented as a function of t d(t) = 300t.
What is the distance?Distance is a numerical representation of the distance between two items or locations. Distance refers to a physical length or an approximation based on other physics or common usage considerations.
It is given that:
An airplane is flying at a speed of 300 mi/h at an altitude of one mile.
As we know,
From the relation between time-distance-speed
distance = time×speed
Speed = 300 mi/h
let distance be d and time be t
d = 300t
We can write the above equation as:
The horizontal distance that the plane has flown:
d(t) = 300t
Thus, the horizontal distance that the plane has flown can be represented as a function of t d(t) = 300t.
The question is incomplete.
The complete question is in the picture, please refer to the attached picture.
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......................................
Answers
In general, a
person is 1% shorter in the evening than in the
morning. Use your height to write a conditional
that uses this fact.
Answers
The conditional statement is that; If generally a person is 1% shorter in the evening than in the morning, then my height is about 1.62326 m in the evening.
How to write conditional statements?
First, we will calculate how much is one percent of your height and then subtract that percentage from the total height to get your" height in the evening".
For example:
If my height is 1.64 m, then;
0.01 * 1.64 m = 0.0164 m
Step 2 is;
1.64m - 0.0164m = 1.62326 m
Now, using these height, let's write a conditional that uses the described fact:
Conditional:
If generally a person is 1% shorter in the evening than in the morning, then my height is about 1.62326 m in the evening.
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Juan and Mahdi are to run a race, but Juan can run 18 meters per second, whereas Mahdi can run 6 meters per second. If Mahdi has a 228 -meter head start, how long will it be before the faster runner catches the slower runner, if they begin at the same time? Round your answer to the nearest whole number.
Answers
Answer:
19 seconds
Step-by-step explanation:
You want to know the time it takes for Juan, running 18 m/s, to overtake Mahdi, running 6 m/s, if Mahdi has a 228 meter head start.
Closing timeThe time to close the gap between the runners will be ...
time = distance/speed
time = (228 m)/(difference in speeds) = (228 m)/(18 m/s -6 m/s)
time = (228 m)/(12 m/s) = (228/12) s = 19 s
It will take Juan 19 seconds to catch Mahdi.
__
Additional comment
The fastest human runner will not exceed 12.3 meters per second, especially over a 300+ meter distance.
Keith works at a garage. In the last week, the garage worked on 105 vehicles.
Out of these 35 were MOTS.
What percentage of the work was MOTS? Give your answer to 2 decimal places.
Answers
In the garage 33.33 percentage of work were MOTS.
Percentage is defined as per-cent, that is out of hundred. The percentage is calculated by dividing the acquired value by the total value and multiplying the result by 100.
Now in the given problem,
Total number of vehicles on which Keith worked is 105.
The number of MOTS among the 105 is 35.
Now we have to calculate the percentage of the MOTS out of a total 105 vehicles.
Let us use the formula=(number of MOTS ÷ total vehicles)×100
Therefore required percentage=[tex]\frac{35}{105}\times 100\%[/tex]
or, percentage=33.333...%
Therefore we can conclude that the total percentage of MOTS were approximately 33.33%(rounded to 2 decimal places).
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A business has a budget of $720 to spend on their marketing campaign. Each month, x, the business spends $40 on different advertisements. The amount of money remaining in the budget
for marketing can be modeled by the function Mx)=-40x+720. Based on the graph of the linear function M(x) and the context of the problem, what is the domain?
O [10, 18]
O [10, 720]
0 [10, 8]
O [R]
Answers
A reasonable domain for this function is [0, 18)
Domain of a functionThe domain of a function is a value of the independent variable of the function for which it exists.
Given the equation below:
M(x)= -40x+720
Since the function represents the The amount of money remaining in the budget for marketing then the function f(x) must be greater than zero;
−40x + 720 > 0
-40x> -720
x < 720/40
x < 18
This shows that a reasonable domain for this function is [0, 18)
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Through (5,8); slope -2
X=-2×+18
How did we get 18
Answers
Answer:
y = -2x + 18
Step-by-step explanation:
Slope m = -2 with y = 8, x = 5
y = mx + b
8 = (-2)(5) + b
8 = -10 + b
b = 8 + 10 = 18
then y = -2x + 18
Find the midpoint of the segment with the given endpoints.
(6,-6) and (1,2)
Answers
The midpoint of the line segment of the endpoints (6,-6) and (1,2) is given as:
(3½,-1)
What is the midpoint of a line segment in geometry?
The midpoint of a line segment is simply the centre point of a give segment that cuts or divides it into two congruent segments.
That is for example, if there are two points with coordinates (x1,y1) and (x2,y2), then their midpoint will be ; [(x1+x2)÷2,(y1+y2)÷2]
From the given question!
x1=6, x2=1 and y1=-6,y2=2
Thus, the midpoint is derived as;
[(6+1)÷2,(-6+2)÷2]
[(7)÷2,(-4)÷2]
(3½,-1).
Find the least numbers which must be subtracted from the following number to make them perfect square
42029
Answers
Step-by-step explanation:
[tex] 3\sqrt{8} + \sqrt{114} [/tex]
Identify the pair(s) of congruent angles in the figure. Explain how you know they are congruent.
by the Congruent Complements Theorem
$\angle NSP\cong\angle QSR$ by the Congruent Complements Theorem
$\angle MSP\cong\angle PSR$ by the Right Angles Congruence Theorem
$\angle MSP\cong\angle PSR$ by the Right Angles Congruence Theorem
$\angle MSN\cong\angle PSQ$ because they have the same measure
$\angle MSN\cong\angle PSQ$ because they have the same measure
$\angle MSN\cong\angle QSR$ by the Vertical Angles Congruence Theorem
$\angle MSN\cong\angle QSR$ by the Vertical Angles Congruence Theorem
$\angle NSP\cong\angle PSQ$ because they have the same measure
$\angle NSP\cong\angle PSQ$ because they have the same measure
$\angle NSQ\cong\angle MSP$ by the Right Angles Congruence Theorem
$\angle NSQ\cong\angle MSP$ by the Right Angles Congruence Theorem
$\angle NSQ\cong\angle PSR$ by the Right Angles Congruence Theorem
Answers
∠NSP ≅ ∠QSR, ∠MSP ≅ ∠PSR, ∠MSN ≅ ∠PSQ, ∠NSQ ≅ ∠MSP, ∠NSQ ≅ ∠PSR are the congruent angles.
∠NSP ≅ ∠QSR by the Congruent Complements Theorem. Because both are complementary to ∠PSQ.
∠MSP ≅ ∠PSR by the Right Angles Congruence Theorem. Because both ∠MSP = ∠PSR = 90°.
∠MSN ≅ ∠PSQ because they have the same measure. ∠MSN = ∠PSQ = 50°. Angles with same measure are congruent to each other.
∠NSQ ≅ ∠MSP by the Right Angles Congruence Theorem. Because ∠NSQ = ∠MSP = 90°.
∠NSQ ≅ ∠PSR by the Right Angles Congruence Theorem. Because ∠NSQ = ∠PSR = 90°.
Congruent Complements theorem states that if two angles are complementary (sum = 90°) to the same angle, then they are congruent.
Right Angles Theorem states that two right angles(90° angles) are congruent to each other.
Vertical Angles Congruence theorem states that two opposite angles formed between intersection of two lines are congruent. Here, ∠MSN is not congruent to ∠QSR because they are not vertical opposite angles.
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What’s the length of AZ. If A is between Y and Z, Y A = 22, AZ = 16x, and Y Z = 166
A) 9
B) 22
C) 144
D)188
Answers
Answer:
c) 144
Step-by-step explanation:
Since A is between Y and Z,
[tex]YZ = YA + AZ[/tex]
[tex]116 = 22 + 16x\\16x = 116 - 22\\16x = 144\\x = 9[/tex]
however, the question asks for [tex]the length of AZ[/tex], not the value of x.
Thus,
[tex]AZ = 16x[/tex]
[tex]AZ = 16\\[/tex] × [tex]9[/tex]
[tex]AZ = 144[/tex]
Find the domain and range of the relation. Also determine whether the relation is a function.
{(6,3), (8,5), (-4,-4), (5,5)}
Answers
The domain is:
D: {-4, 5, 6, 8}
The range is:
R: {-4, 3, 5}
And yes, the relation is a function.
How to determine the domain and range?
For a relation that maps elements x into elements y (in the form (x, y)), we define the domain as the set of the inputs (values of x) and the range as the set of the outputs (values of y).
Here our relation is defined by: {(6,3), (8,5), (-4,-4), (5,5)}
The domain is the set of the first values of each pair, so we have:
D: {-4, 5, 6, 8}
The range is the set of the second values of each pair:
R: {-4, 3, 5}
Now, is this a function?
Yes, it is a function, because each input is mapped into only one output.
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Find the solutions to x² = 28.
A. x=+7√2
OB. x=+2√/14
OC. x=+14√2
X
OD. x=+2√7
X
Answers
Answer:
x=+2√7
Step-by-step explanation:
Please help I’m having trouble
Answers
12x7x5=420
Answer 2 and 3. I’ll give Brainlyest and a lot of points.
Answers
Answer:
Please add the graph for these problems so I can solve your questions? Thank you!
Step-by-step explanation:
Amy printed a square picture that was 225% the size of the original picture. The length of the side of her original picture was 4 inches. What is the length of the side of Amy’s printed picture
Answers
The length of the side of Amy’s printed picture is 6 inches.
What is defined as a square?A square is a two-dimensional closed shape with four equal sides.
A square is made up of four sides and four vertices.A square's sides are all the same length.All interior angles are identical and correct.The total of all interior angles is 360°.Now, as per the given question;
The original length of the side of the square was 4 inches.
Thus, the area of the old photo was;
A = side²
A = 4²
A = 16 square inch.
Now, as the picture is increased by 225%.
Then, increase the area by 225%.
New area = 225% of old area
= (16×225)/100
New area = 36
The length of the side of the new photograph is;
New area = length²
length = √New area
length = √36
length = 6 inches.
Therefore, the length of the side of Amy’s printed picture is 6 inches.
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2. Which statement correctly describes the
perpendicular bisector construction below?
Answers
The statement correctly describes the perpendicular bisector construction below The construction of JK to bisect GH: Option D is the correct answer
What is perpendicular bisector?The term perpendicular bisector refers to the line that divides another line into two equal parts, making an angle of 90 degrees. The term is made up of two words, perpendicular and bisector.
Perpendicular refers to angle 90 degrees, while the bisector refers to dividing into two equal parts.
In the picture, the main line is GH while JK is the line constructed to bisect the main line
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