Simplify the function f(x)= 1/2 (27) 2x/3. Then determine the key aspects of the function
Answers
[tex]f(x)= \frac{1}{2}(27)^{ \frac{2x}{3} } \\ \\ f(x)= \frac{1}{2}((27)^{ \frac{2}{3} })^{x} \\ \\ f(x)= \frac{1}{2}(9)^{x} [/tex]
From the above equation, we can answer all the 4 questions.
a) Initial value occurs when x = 0
So,
[tex]f(0)= \frac{1}{2}(9)^{0}= \frac{1}{2} [/tex]
So, the Initial value of the function is 1/2.
b) Simplified Base:
The base of an exponential function is the term which contains the exponent.
From the simplified equation, we can see that the base has been simplified and is equal to 9.
So, the simplified base is 9
c) Domain:
From the equation we can see that there is no such value which makes the function undefined. So all reall numbers are included in the Domain.
So, Domain is Set of All real numbers.
In Interval notation this can be expressed as: (-∞,∞)
d) Range:
Range is the set of all y values that a function can take. Since the graph is not shifted vertically up or down, the range will be the set of all real numbers greater than zero.
So, Range is equal to All real numbers greater than 0.
In Interval notation, this can be expressed as (0,∞)
Answer:
The last one is y>0
Related Questions
The sum of two numbers is 25, and the difference of their squares is 75. what are the two numbers?
Answers
14*14=196
11*11=121
196-121=75
the answer is x=14 and y=11
The points at which a quadratic equation intersects the x-axis are referred to as
Answers
There is a number of different things these are referred to as. The first of which is the zeroes of the equation. They are called the zeroes of the equation due to the fact that this is where the function (or y-value) is equal to 0.
It can also be referred to as the x-intercepts. This is because it is where the graph intercepts the x-axis.
The points at which a quadratic equation intersects the x-axis are referred to as x intercepts or zeros or roots of quadratic equation
Given :
The points at which a quadratic equation intersects the x-axis
The points at which the any quadratic equation crosses or touches the x axis are called as x intercepts.
At x intercepts the value of y is 0.
So , the points at which a quadratic equation intersects the x-axis is also called as zeros or roots of the quadratic equation .
The points at which a quadratic equation intersects the x-axis are referred to as x intercepts or zeros or roots of quadratic equation
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A student ate 3/20 of all candies and another 1.2 lb. Another student ate 3/5 of the candies and the remaining 0.3 lb. Altogether, what weight of candies did they eat?
Answers
x-------> total weight of candies
we know that
x=(3/20)*x+1.2+(3/5)*x+0.3
x=(3/20)*x+(3/5)*x+1.5----> multiply by 20----> 20x=3x+12x+30
20x=15x+30
20x-15x=30
5x=30
x=6 lb
the answer is
6 lb
If matrix A is a 6 × 5 matrix, which order of matrix can be multiplied by matrix A?
Answers
Answer:
5 x n, where n is any natural number.
Step-by-step explanation:
If A is a 6 x 5 matrix, it means that A has 6 rows and 5 columns, this means that if a matrix B can be multiplied by A, then each one of its columns must have 5 entries. Therefore, B has to be a matrix with 5 rows and it might have any arbitrary number of columns, in other words B is a 5 x n matrix, where n must be a natural number bigger than 1.
Javier is 175% heavier than his brother . if javier brother weigh 80 pounds how much do javier weigh
Answers
1% = 80 ÷ 100 = 0.8 pound
175% = 0.8 x 175 = 140 pounds
Answer: Javier weighs 140 pounds.
A sound wave travels through iron at a rate of 5120 m/s. At what rate do the sound waves travel at km/h
Answers
Answer:
Sound waves travel through iron at 11,427,840 km/h.
Step-by-step explanation:
A kilometer is equal to 0.62 miles. An hour is equal to 3600 seconds. Therefore, if a sound wave travels through iron at a rate of 5120 m/s, it would be equal to (5120 x 0.62) km/s. This is equal to 3174.4 km/s which, multiplied by 3600 (the number of seconds that make an hour), would determine the speed in which a sound wave travels through iron. So, as 3174.4 x 3600 is 11,427,840, so sound waves travel at 11,427,840 km/h.
Let v1 = (-6,4) and v^2 = (-3,6). Compute the following
V^1 * V^2
Answers
Find the value of n such that x2-10x+n is a perfect square trinomial.
A. -5
B. 50
C. 100
D. 25
Answers
D. 25
The explanation is shown below:
1. You have that:
(a ± b)^2=a^2 ± 2ab + b^2
2. The expression given in the problem is:
x^2-10x+n
Where x^2=a and 2ab=10x
2b=10
b=10/2
b=5
3. Therefore, you have:
b^2=5
b^2=25
b^2=n
n=25
Answer:
the answer is 25!!
Step-by-step explanation:
I hope this is helpful to sum of y'all!!
your mom is looking at he amount of sugar in two different drinks the apple juice has 216 grams sugar for every 8 serving the sprite 234 grams of sugar for every 9 serving which drink has more sugar
Answers
The longer leg of a right triangle is 4 more than twice the shorter leg. The hypotenuse is 4 less than three times the short leg. What is the value of the hypotenuse?
Answers
shorter leg =x
longer leg = 2x + 4
hypotenuse = 3x - 4
Given that a² + b² = c²
x² + (2x + 4)² = (3x - 4)²
Remove brackets : (a + b)² = a² + 2ab + b²:
x² + 4x² + 16x + 16 = 9x² - 24x + 16
Moved all terms to the left:
x² + 4x² + 16x + 16 - 9x² + 24x - 16 = 0
Combine like terms:
-5x² + 40x = 0
Multiply all terms by -1:
5x² - 40x = 0
Take out 5x as common factor:
5x(x - 8) = 0
Apply zero product property:
5x = 0 or x - 8 = 0
Evaluate:
x = 0 or x = 8
Since length cannot be zero, shorter leg = 8
shorter leg = 8
Hypotenuse = 3(8) - 4 = 20
Answer: 20 units
The graph shows a probability distribution.
What is P(1.5≤X≤4.5)?
Enter your answer, as a decimal
P(1.5≤X≤4.5) =
Answers
In the picture, the area that colored red is from x=5 till x=1, so the length would be: 5-1= 4 blocks. The height of all the block is the same, which is p=1. The total area of the red block would be: length * height = (5-1) * 1 = 4
The area of P(1.5≤X≤4.5) would be: length * height = (4.5-1.5) * 1= 3
The probability of P(1.5≤X≤4.5) would be: area/ total area= 3/4= 0.75
find the point slope equation for (-2,-20) and (9,79)
Answers
x1: -2
x2: 9
y1: -20
y2: 79
The general form for writing a point slope equation is y-y1 = m(x-x1)
M is your slope.
To find the slope, plug the values you just identified into the equation y2-y1/x2-x1
This leaves you with 79+20/9+2
Which equals 9, so your slope (m) equals nine.
you can now plug everything into the general form which gives you:
y+20 = 9(x+2)
The point-slope equation for the points (-2,-20) and (9,79) is y + 20 = 9(x + 2).
Explanation:To find the point-slope equation, we can use the formula: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Firstly, we calculate the slope using the formula: m = (y2 - y1)/(x2 - x1). Given the points (-2, -20) and (9, 79), the slope is (79 - (-20))/(9 - (-2)). Simplifying the expression, we get m = 99/11 = 9.
Next, we substitute one of the points and the calculated slope into the point-slope formula. Let's choose (-2, -20). So the equation becomes: y - (-20) = 9(x - (-2)). Simplifying further, we have: y + 20 = 9(x + 2).
This is the point-slope equation for the given points: y + 20 = 9(x + 2).
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You bought a guitar 6 years ago for $400. It's value decreases by about 13% per year. Write a formula to show the value of the guitar over those six years and determine its current value
Answers
which is value = Original price x (1-percent change)^time
using the information from the problem
Value = 400 x(1-0.13)^6
value = 400 x 0.87^6
value = $173.45
Final answer:
The current value of the guitar after depreciating by 13% per year over 6 years is calculated using the exponential decay formula [tex]V = P(1 - r)^t[/tex], which gives us an approximate current value of $173.45
Explanation:
The given formula represents depreciation, where the value of an asset decreases over time. Applying this to the provided scenario, with an original price of $400 and a 13% annual depreciation rate, we calculate the value after 6 years. Substituting these values into the formula, we find: Value = [tex]400 \times (1 - 0.13)^6 = 400 \times 0.87^6 = \$173.45.[/tex]Therefore, after 6 years, the asset's value is approximately $173.45. Depreciation formulas like this one help assess the declining worth of assets over time due to various factors.
Out of the total respondents, the percentage of respondents from the 46–55 age group who rated the film excellent is
%. Write your answer up to two decimal places.
NextReset
Answers
The percentage of respondents from the 46-55 age group who rated the film excellent is 20 %.
To determine the percentage of respondents from the 46-55 age group who rated the film excellent, you need two pieces of information: the number of respondents in that age group who rated the film excellent and the total number of respondents in that age group. The formula for calculating the percentage is:
[tex]\[ \text{Percentage} = \left( \frac{\text{Number of respondents who rated excellent}}{\text{Total number of respondents in the age group}} \right) \times 100 \][/tex]
Assuming you have these numbers, let's denote the number of respondents who rated the film excellent in the 46-55 age group as X and the total number of respondents in that age group as Y. The formula becomes:
[tex]\[ \text{Percentage} = \left( \frac{X}{Y} \right) \times 100 \][/tex]
For example, if 20 respondents in the 46-55 age group rated the film excellent out of a total of 100 respondents in that age group, the calculation would be:
[tex]\[ \text{Percentage} = \left( \frac{20}{100} \right) \times 100 = 20\% \][/tex]
Therefore, replace X and Y with the actual numbers from your data to find the specific percentage.
Classify each number as rational or irrational.
1. 0.329
2. 127.5
3. -89
4. √--- 101
Answers
127.5 : Rational numbers
-89 : Rational numbers
[tex] \sqrt{101}[/tex] : Irrational numbers
Hope this helps. - M
Answer:
A rational number is a number with a finite number of numbers after the decimal point, or the numbers after the decimal point repeat a given patern.
An irrational number is a number with infinite numbers afther the decimal point, and those numbers have no pattern, are all "random", some examples are the roots of prime numbers, like square root of 2.
1) 0.329
It has a finite quantity of numbers after the decimal point: rational
2) 127.5
It has a finite quantity of numbers after the decimal point: rational
3) -89
It has a finite quantity of numbers after the decimal point: rational
4) [tex]\sqrt{101[/tex]
101 is a prime number, so the square root of 101 must be a irrational number.
How many Liters of a liquid can be held in a barrel
Answers
how many times greater is the value of the 2 in 204,936 than the value of the 2 in 124,936
A. 1/100
B. 1/10
C. 10
D. 100
E. 1,000
Answers
Summer spent $55.60 on clothes before sales tax. If sales tax is 7.5%, what did Summer spend with tax included? *Add Explanation If U Can*
Answers
A parabolic archway is 12 meters high at the vertex. at a height of 10 meters, the width of the archway is 8 meters. how wide is the archway at ground level?
Answers
1. You have that the parabola is 12 meters high at the vertex and at a height of 10 meters, the width is 8 meter.
2. Therefore, you have:
x^2=-4a(y^2-k)
4^2=-4a(10-12)
16=-40a+48
16=8a
a=16/8
a=2
y=0
x^2=-8(0-12)
x^2=96
x=√96
x=9.79x2
x=19.58
3. Therefore, as you can see, the answer is: 19.58 meters.
We found the width at ground level to be approximately 19.6 meters.
We start by assuming the equation of the parabola to be of the form y = ax² + bx + c.
Since the vertex of the parabolic arch is at (0, 12), we know that c = 12. Next, we use the condition that at y = 10 meters, the width is 8 meters, meaning the points (4, 10) and (-4, 10) lie on the parabola. Plugging these into the equation:
For (4,10): 10 = a(4)² + b(4) + 1210 = 16a + 4b + 1216a + 4b = -2 (1)Solving for the same with (-4,10) is unnecessary as it results in the same equation. Next, we use the vertex form y = a(x - h)² + k ⇨ y = a(x - 0)² + 12 ⇨ y = ax² + 12.
From equation (1), we have 16a + 4b = -2. Since the vertex form tells us there is no 'b' term in the simplest form, 'b' must be 0. Thus, 16a = -2 ⇨ a = -1/8.
The equation of the parabola is therefore y = -(1/8)x² + 12.
To find the width at ground level (y = 0):
0 = -(1/8)x² + 12(1/8)x² = 12x² = 96 ⇨ x = ±√96 = ±4√6 ≈ ±9.8 meters.Thus, the total width of the arch at ground level is approximately 2 × 9.8 = 19.6 meters.
Twelve added to a number equals 18 find the number
Answers
x = 18, by subtracting 'x' from both sides, and adding 8 to both sides.
Which points in the scatter plot are outliers?
Point A
Point F
Point H
Point K
Point M
Answers
If you draw a line of best fit, you'll see that only two points lie way outside the line. These points are points K and F.
Hope this helps!! :)
If there's anything else that I can help you with, please let me know!
Answer:
the points are k and f
Step-by-step explanation:
Find the probability of drawing a heart or a black card from a standard deck of cards
Answers
Hearts and black cards make up 3 of the 4 suits of cards. You could count the amount of them and then divide by 52 cards, or you can divide 3 by 4 since there are the same amount in every suit.
The probability of drawing a heart or a black card from a standard deck of cards is 3/4.
Explanation:In a standard deck of cards, there are 52 cards in total. However, since we are interested in finding the probability of drawing a heart or a black card, we need to determine how many cards satisfy this condition.
There are 13 hearts in a deck, so the probability of drawing a heart is 13/52 or 1/4. Additionally, there are 26 black cards (13 spades and 13 clubs) in a deck, so the probability of drawing a black card is 26/52 or 1/2.
To find the probability of drawing a heart or a black card, we can add the probabilities together: 1/4 + 1/2 = 3/4.
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If nine people are in a room and every person shakes hands exactly once with each of the other people, how many handshakes will occur
Answers
Answer: 36 handshakes
Crater Lake Oregon is a roughly circular lake. The lake basin formed about 7000 years ago when the top of a volcano exploded in an immense explosion. Hillman Peak, Garfield Peak, and Cloudcap are three mountain peaks on the rim of the lake. The peaks are located in a coordinate plane at H(-4,1), G(-2,-3), and C(5,-2). Find the coordinates of the center of the lake.
Answers
(Xa) ² + (Yb) ² = r²
They do not give us the center or the radio, but we have 3 points.
With the points H (-4,1), G (-2, -3) and C (5, -2) we find 3 equations
1) (-4-a) ² + (1-b) ² = r²
2) (-2-a) ² + (-3-b) ² = r ²
3) (5-a) ² + (-2-b) ² = r ²
Now we have a system of 3 equations with 3 unknowns: r, a, b
When solving the system we have to:
r = 5
a = 1
b = 1
Therefore, the center of the circumference of the lake is the point (a, b)
That is, the point (1,1)
Answer:
The coordinates of the center of the lake are [tex](\frac{-1}{3},\frac{-4}{3})[/tex].
Step-by-step explanation:
It is given that Hillman Peak, Garfield Peak, and Cloudcap are three mountain peaks on the rim of the lake. The peaks are located in a coordinate plane at H(-4,1), G(-2,-3), and C(5,-2).
If we joint these points, then we get a triangle and the center of a triangle is known as centroid.
The formula for centroid of a triangle is
[tex]Centroid=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})[/tex]
The centroid of the triangle H(-4,1), G(-2,-3), and C(5,-2) is
[tex]Centroid=(\frac{-4-2+5}{3},\frac{1-3-2}{3})[/tex]
[tex]Centroid=(\frac{-1}{3},\frac{-4}{3})[/tex]
Therefore the coordinates of the center of the lake are [tex](\frac{-1}{3},\frac{-4}{3})[/tex].
Using sin^2x+ cos^2 x=1 and other trig properties, simplify the following: (cos^2Ø*tan^2Ø)/(1-cos^2 Ø.
Answers
[tex]Used:\\\\\sin^2\O+\cos^2\O=1\to1-\cos^2\O=\sin^2\O\\\\\tan\O=\dfrac{\sin\O}{\cos\O}[/tex]
Select all that are undefined. sec (45°) csc (0°) sec (90°)
Answers
csc(0°) = 1/sin(0°) = 1/0 /undefined/
sec(90°) = 1/cos(90°) = 1/0 /undefined/
Final answer:
The sec(45°) and sec(90°) are defined, but csc(0°) is undefined.
Explanation:
To determine which of the given trigonometric functions are undefined, we need to understand the domains of these functions. In trigonometry, the secant and cosecant functions are undefined for angles where the cosine and sine functions are equal to zero, respectively.
For sec(45°), the cosine of 45° is equal to √2/2, which is not zero. Therefore, sec(45°) is defined.For csc(0°), the sine of 0° is equal to 0, which results in division by zero. Hence, csc(0°) is undefined.For sec(90°), the cosine of 90° is equal to 0, which again leads to division by zero. Thus, sec(90°) is undefined.Which of the following is equal to 7÷9/5
A: 9×7/5 B: 7×9/5 C: 9×5/7
D: 7×5/9
Answers
Final answer:
To find the value of 7÷9/5, we multiply 7 by the reciprocal of 9/5, which is 5/9, resulting in the expression 7×5/9. The correct answer is D: 7×5/9.
Explanation:
To solve the expression 7÷9/5, we must interpret the division as multiplication by the reciprocal. In other words, dividing by a fraction is the same as multiplying by its reciprocal. Therefore, 7 divided by 9/5 is equal to 7 multiplied by the reciprocal of 9/5, which is 5/9.
So, the calculation will be 7×5/9. Here's the math:
Identify the reciprocal of 9/5, which is 5/9.
Multiply 7 by the reciprocal of 9/5: 7 × 5/9.
Calculate the product: 7 × 5 = 35, and 35 ÷ 9 does not simplify further in this context.
The correct answer from the options provided is D: 7×5/9. The other options, A: 9×7/5, B: 7×9/5, and C: 9×5/7 do not represent the correct transformation from division to multiplication by the reciprocal.
A swimming pool has four faucets. the first can fill the entire pool with water in two days, the second - in three days, the third - in four days, and the last one can fill the pool in 6 hours. how long will it take to fill the pool using all 4 faucets together (be exact)?
Answers
A-2 days
B-3 days
C-4 days
D-6 hours=6/24=1/4 days
Thus the time taken for the 4 faucets working together to fill the pool will be the GCD of the time taken by the 4, which will be:
GCD(2,3,4,0.25)
=1
=1 day
Answer: 1 day or 24 hours
A french fry stand at the fair serves their fries in paper cones. The cones have a radius of 222 inches and a height of 666 inches. It is a challenge to fill the narrow cones with their long fries. They want to use new cones that have the same volume as their existing cones but a larger radius of 444 inches. What will the height of the new cones be?
Answers
To find the height of the new cone with a larger radius that maintains the same volume, we calculate the volume of the original cone and solve for the new height using the cone volume formula.
Explanation:The student wants to find out the new height of the cone with a larger radius while maintaining the same volume as the original cone. The volume of a cone is given by the formula V = 1/3 πr²h. To find the height of the new cone, we'll use the volume of the original cone as the constant value.
Let's first calculate the volume of the original cone:
V_original = 1/3 π(222²)(666)Now, let's set up an equation with the new radius to solve for the new height (h_new):
1/3 π(444²)(h_new) = V_originalh_new = (V_original) / (1/3 π(444²))By substituting the V_original from step 1 into step 3, we can solve for h_new. The result will be the height of the new cone with a radius of 444 inches that has the same volume as the original cone.
How do you combine like terms of #-10+7x+24-2x#?