The ordered pairs (1, 6), (2, 36), (3, 216), (4, 1296), and (5, 7776) represent a function. What is a rule that represents this function?
Answer:
The rule is [tex]\\ f(x) = 6^{x}[/tex]
Step-by-step explanation:
The resulting values are powers of 6:
[tex]\\ 6^{1} = 1[/tex]
[tex]\\ 6^{2} = 36[/tex]
[tex]\\ 6^{3} = 216[/tex]
[tex]\\ 6^{4} = 1296[/tex]
[tex]\\ 6^{5} = 7776[/tex]
One factor that is not defined in the question is whether the domain (the set of values for which a function is defined [WolframAlpha, 2019]) of the function [tex]\\ f(x) = 6^{x}[/tex] is only for x = {1, 2, 3, 4, 5} or defined for all possible values this function can take, that is, x = (-∞;∞) or all real numbers.
As you can see from the graph attached:
Theoretically, this function could take all values for x = (-∞;∞). If this were the case, the function is a continuous line with resulting values (that is, the range of the function) for f(x)>0 (0;∞). The graph below shows only positive values (x > 0) and values for x ≤ 5.For values x = {1, 2, 3, 4, 5}, the function is discrete (not a continuous line) and only defined for these values. In this case, the function would be represented only by the points in the graph, and the values it could take would be only f(x) = {6, 36, 216, 1296, 7776}, which would represent the range for this function.In the last case, we have to redefine the function as:
[tex]\\ f(x) = 6^{x}[/tex] for [tex] x = \{1, 2, 3, 4, 5\}[/tex].
If x2 − 8xy + 12y2 = 8 and x − 6y = 3, then x − 2y =
the area of a rectangle is 15x^2y^5. if the length is 5xy^3, what is the width?
Two less than three times the width of a rectangle is equal to the length. The area of the rectangle is 65 square ft. What is the length of the rectangle?
Answer:
Length of the rectangle = 13 ft
Step-by-step explanation:
Let l be the length and w be the width of this rectangle.
Two less than three times the width of a rectangle is equal to the length, that is
3w - 2 = l---------------------------eqn 1
The area of the rectangle is 65 square ft
l x w = 65
Substituting eqn 1
(3w - 2) x w = 65
3w² - 2w -65 = 0
Solving quadratic eqn
[tex]w=\frac{-(-2)\pm \sqrt{(-2)^2-4\times 3\times (-65)}}{2\times 3}=\frac{2\pm \sqrt{4+780}}{6}\\\\w=\frac{2\pm \sqrt{784}}{6}\\\\w=\frac{2\pm 28}{6}\\\\w=\frac{2+28}{6}\texttt{ or }w=\frac{2-28}{6}\\\\w=5\texttt{ or }w=-4[/tex]
w cannot be negative
Hence width, w = 5 ft
Substituting in eqn 1
3 x 5 - 2 = l
l = 13 ft
Length = 13 ft
Length of the rectangle = 13 ft
Does this set of ordered pairs represent a function? Why or why not?
{(–2, 2), (–1, 2), (3, –1), (3, 1), (4, 11)}
A. Yes, because there are two x-values that are the same.
B. No, because two of the y-values are the same.
C. Yes, because every x-value corresponds to exactly one y-value.
D. No, because one x-value corresponds to two different y-values.
Consider the functions f(x) = x2 − 13 and g(x) = x + 5. What is the value of (f − g)(−4)? ...?
Answer- The Answer is -12
Step-by-step explanation:
I took the test
The table contains data on the mileage displayed on a bicycle odometer as it relates to the number of days since the bicycle was purchased. Using the Scatter Plot tool, find the value of the correlation coefficient for this data set.
0.8
1.0
0.3
0.1
Answer:
Hence, the correlation coefficient is:
r=1.0
Step-by-step explanation:
We know that the correlation coefficient ''r'' is calculated by the formula:
[tex]r=\dfrac{\sum{XY}}{\sqrt{\sum{X^2}\sum{Y^2}}}--------------(1)[/tex]
Let x denote the data point (Number of days since purchase)
Let y denote the data point (Mileage Displayed on odometer)
x y X=x-x' Y=y-y' XY X² Y²
15 67 -20 -95 1900 400 9025
25 122 -10 -40 400 100 1600
35 164 0 2 0 0 4
45 210 10 48 480 100 2304
55 247 20 85 1700 400 7225
x' be the mean of the data of the x-values.
[tex]x'=\dfrac{15+25+35+45+55}{5}\\\\\\x'=\dfrac{175}{5}\\\\\\x'=35[/tex]
Also let y' denote the mean of the y-values.
[tex]y'=\dfrac{67+122+164+210+247}{5}\\\\\\y'=\dfrac{810}{5}\\\\\\y'=162[/tex]
Now we have:
∑ XY=4480
∑ X²=1000
∑ Y²=20158
Hence, we put all the values in the formula (1) to obtain:
r=0.997 which is close to 1.0
Also from the scatter plot we could observe that the relationship is linear and also strong positive relationship.
Hence, the correlation coefficient is 1.0
What number represents the same amount as 2 hundreds 14 tens 3 ones
The representation of the given number will be given below such that it sum should be 343.
What is summation?A summation, also abbreviated as a sum, is the outcome of adding two or more numbers or quantities. Here are always an integer number of terms in a summation.
There might simply be two terms, but there might also be 100,000 or a million. Summations exist with an infinite number of parameters.
The summation is to add or subtract any two or more terms by doing the operation.
The representation of 200, 14, and 3 will be
200 hundred
14 tens
+ 3 ones
--------------------
343
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Any ordered pair that makes all equations in a system of equations true is a(n) ...?
The cost of a service call to fix a washing machine can be expressed by the linear function y=45x+35 where y represents the total cost and x represents the number of hours it takes to fix the machine. What does the slope represent?
a-the cost for each hour it takes to repair the machine
b=the cost for coming to look at the machine
c-the total cost for fixing the washing machine
d-the amount of time that it takes to arrive at the home to make the repairs
Answer:
The slope represents the cost for each hour it takes to repair the machine.
Step-by-step explanation:
We are given that the cost of a service call to fix a washing machine can be expressed by the linear function
[tex]y=45x+35[/tex]
Where
y=Total cost of a service to fix the machine.
x=Number of hours takes to fix the machine
We have to find that slope in this function what represents.
General equation of line is given by
[tex]y=mx+c[/tex]
Where m=Slope of line
c=y- intercept
Comparing the general equation with the given function
We get m=45,c=35
m=45 represents the cost for each hour it takes to repair the machine because when number of hours multiplied by 45 then we get total cost of x hours.
Hence, the slope represents the cost for each hour it takes to repair the machine.
how do i work out sinx+2sinxcosx=0 in radians?
Electrical energy is converted into which three forms of energy by a hair dryer?
A hair dryer converts electrical energy mainly into thermal energy to heat the air, but it also produces mechanical energy to power a fan and creates sound energy during operation.
A hair dryer converts electrical energy into three main forms of energy: thermal energy, mechanical energy, and sound energy. When a hair dryer is turned on, the electrical energy is primarily converted into thermal energy to heat the air. This is achieved by passing the electric current through resistive heating coils, which transfer energy to the surrounding air as heat. Furthermore, some of the electrical energy is transformed into mechanical energy to power a fan that blows the hot air out of the dryer. Lastly, the operation of the hair dryer also produces sound energy due to the movement of the fan and the rushing air.
tan^4x-sec^4x=1-2 sec^2(x)
verify
An 18 m tall tree is broken during a severe storm. The distance from the base of the trunk to the point where the tip touches the ground is 12 m. At what height did the tree break?
Translate the phrase into a math expression.
Twenty divided by the sum of 4 and 1.
(1+4) ÷ 20
20 ÷ (1+4)
1 + (4÷20)
(20÷4) + 1
Answer:
[tex]\displaystyle 20 \div (1 + 4)[/tex]
Explanation:
It is EXACTLY how it sounds, and of course, you must include grouping symbols to separate both parts of the expression.
I am joyous to assist you anytime.
"If a parallelogram has four right angles, then it is a rectangle." What is the inverse of this conditional statement? A. If a parallelogram has four right angles, then it is a rectangle. B. If a parallelogram does not have four right angles, then it is a rectangle. C. If a parallelogram has four right angles, then it is not a rectangle. D. If a parallelogram does not have four right angles, then it is not a rectangle.
Answer:
If a parallelogram does not have four right angles, then it is not a rectangle.
Step-by-step explanation:
"If a parallelogram has four right angles, then it is a rectangle."
To get the inverse of the conditional statement, we take the negation of both the hypothesis and the conclusion.
So, the inverse of the given statement will be :
If a parallelogram does not have four right angles, then it is not a rectangle.
"The graph below shows the population of mice in an ecosystem where the mice are not allowed to enter or leave.
Which best describes the population at point A?
A.The death rate is higher than the birthrate.
B.The population is decreasing.
C.The birthrate is higher than the death rate.
D.It has reached carrying capacity"
The point marked A is the point where the birth rate is higher than the death rate.
What is birth rate/ death rate?The term birth rate refers to the rate at which new organims are introduced into the population and the death rate refers to the rate at which individuals exit in the population.
In this case, when we observe the graph, we can see that the point marked A is the point where the birth rate is higher than the death rate.
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8 foot piece of wood cut into how many 7/8 inches?
The graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.
True
False
Answer:
true
Step-by-step explanation:
The statement 'The graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.' is True.
What s graph of a function?"It is a set of points that satisfy given function."
What is transformation of a function?"It means that the curve representing the graph either 'moves to left / right/ up/ down' or 'it expands or compresses' or 'it reflects'."
What is translation of function?"A shift of the graph of a function up, down, left, or right, without changing the shape, size, or dimensions of the graph."
For given question,
We have been given a function y = 6cos x - 3
Also we have been given a transformed function y = 6cos(x - 2) - 3
We know that when function f(x) is transformed to f(x - k) then the graph of f(x) is translated 'k' units right.
So, the graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.
Therefore, the statement 'The graph of y = 6cos(x - 2) - 3 is obtained by shifting the graph of y = 6cos x - 3 horizontally 2 units to the right.' is True.
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a cheese pizza cost $8.75 each additional cost topping costs $1.25. if you order one pizza and have $12.50. how many toppings can you have on the pizza.
the sum of two numbers is 85. twice one number plus four times the other is 218. find the numbers
what is the solution set of the equation?
8/x+2=(x+4)/(x-6)
a. {-12,4}
b. {12}
c. {4,6}
d. {-4,12}
Geometry question help
15 pts will award brainliest if explained
Answer if you understand it only
Can someone help me...again?
Tom gets $12 off a box of chocolates that had an original price of $48. What percentage is the discount ?
how to estimate 184÷9 in multiplication
How do you factorise xy zy?
you are driving at a constant velocity of 100 km per hour. the first landmark you see is your favorite restaurant which happens to be 10 km from your house. which sequence describes the distances you cover at intervals of one hour, starting when you pass the restaurant?
Find the measure of an angle whose measure is 18 degrees less than one-half of the measure of its complement.
If x represents the measure of the angle, then which of the following expressions would equal x?
90 - (0.5x - 18)
0.5(90 - x - 18)
0.5(90 - x) - 18
Answer:
The answer is C, 0.5(90 - x) -18
Step-by-step explanation:
The graphs of the function f (given in blue) and g (given in red) are plotted above. Suppose that u(x)=f(x)g(x) and v(x)=f(x)/g(x). Find each of the following:
u'(1) =
v'(1) =
This is a problem of calculus where the product and quotient rules are used to find the derivatives of functions u and v at x=1. The values need to be taken from the provided graph.
Explanation:This is a calculus question related to the product rule and the quotient rule of derivatives. First, let's explore the definitions: the product rule states that the derivative of the product of two functions is the derivative of the first times the second plus the first times the derivative of the second. The quotient rule states that the derivative of the quotient of two functions is the bottom times the derivative of the top minus the top times the derivative of the bottom, all over the bottom squared. From the graph, we have to find out the values of f(1), g(1), f'(1), and g'(1).
Then, we can calculate u'(1) by using the product rule: u'(1) = f'(1)g(1) + f(1)g'(1). In the same way, we can calculate v'(1) by using the quotient rule: v'(1) = [g(1)f'(1) - f(1)g'(1)] / [g(1)]^2. The specific values need to be derived from the provided graph
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(u’(1) = 5)
[tex](v’(1) = \frac{-13}{4})[/tex]
We have two functions:
(u(x) = f(x)g(x))
[tex](v(x) = \frac{f(x)}{g(x)})[/tex]
We need to find the derivatives of both (u(x)) and (v(x)) at (x = 1).
1. Derivative of (u(x)):
Using the product rule, we have:
[ u’(x) = [tex]v \cdot \frac{du}{dx} + u \cdot \frac{dv}{dx} ][/tex]
Where:
(v = g(x))
[tex](\frac{du}{dx})[/tex] is the derivative of (f(x))
[tex](\frac{dv}{dx})[/tex] is the derivative of (g(x))
From the graph:
At (x = 1), we have (f(1) = 3) and an estimated slope of the tangent line for (f’(1)[tex]\approx -2).[/tex]
Similarly, at (x = 1), we have (g(1) = 2) and an estimated slope of the tangent line for [tex](g’(1) \approx 3).[/tex]
Now let’s calculate (u’(1)):
[ u’(1) = [tex]g(1) \cdot f’(1) + f(1) \cdot g’(1)[/tex] = (2)(-2) + (3)(3) = 5 ]
2. Derivative of (v(x)):
Using the quotient rule, we have:
[ v’(x) = [tex]\frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} ][/tex]
Substitute the values we found earlier:
(v = g(x) = 2)
[tex](\frac{du}{dx}[/tex] = f’(x) = -2)
[tex](\frac{dv}{dx} = g’(x) = 3)[/tex]
Now let’s calculate (v’(1)):
[tex][ v’(1) = \frac{(2)(-2) - (3)(3)}{2^2} = \frac{-13}{4} ][/tex]
Therefore:
(u’(1) = 5)
[tex](v’(1) = \frac{-13}{4})[/tex]