Answer:
use desmos or they work pretty well
The solution of the inequality is -4<x≤1 where x≠-1
What is Inequality?A relationship between two expressions or values that are not equal to each other is called 'inequality.
The given inequality is x²-1/x²+5x+4 ≤0
Let us factor the terms in inequality to find the solution
(x+1)(x-1)/(x+1)(x+4) ≤0
(x-1)/(x+4) ≤0
Find the critical values where the inequality changes sign
Interval 1: x < -4
Interval 2: -4 < x < 1
Interval 3: x > 1
Test each interval to see where the inequality is true or false
For interval 1, we can choose x = -5 as a test point:
(x-1)/(x+4) = (-5-1)/(-5+4) = -6 < 0
Since the inequality is true for x = -5, it is also true for all x < -4.
For interval 2, we can choose x = 0 as a test point:
(x-1)/(x+4) = (0-1)/(0+4) = -1/4 < 0
Since the inequality is true for x = 0, it is also true for all -4 < x < 1.
For interval 3, we can choose x = 2 as a test point:
(x-1)/(x+4) = (2-1)/(2+4) = 1/6 > 0
Since the inequality is false for x = 2, it is also false for all x > 1.
Combine the intervals where the inequality is true -4<x≤1 where x≠-1 is the solution
Hence, the solution of the inequality is -4<x≤1 where x≠-1
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how would the baked loaves be different if you increased the shortening in the french bread formula (p. 135) to 7%?
Increasing the shortening in the French bread formula to 7% would result in a softer and more tender crumb due to the added fat.
French bread is typically made with a small amount of fat, such as butter or oil, to enhance its flavor and texture. However, increasing the shortening to 7% would significantly increase the amount of fat in the formula, which would have several effects on the final product.
Firstly, the added fat would make the dough more extensible, resulting in a softer and more tender crumb. This is because the fat coats the gluten strands in the dough, reducing their ability to form a strong network and resulting in a more open and airy crumb.
Additionally, the added fat would also make the crust softer and less crispy. This is because the fat prevents the crust from drying out and developing a hard, crispy texture.
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Label the net for the cylinder. Then find the surface area of the cylinder. Give your answer in terms of π and as a decimal number rounded to the nearest tenth.
The surface area of the cylinder is approximately 94.2 ft².
What is surface area?Surface area refers to the total area of the external or outer part of an object. It is the sum of the areas of all the individual surfaces or faces of the object. Surface area is typically measured in square units, such as square inches (in²), square feet (ft²), or square meters (m²), depending on the unit of measurement used.
According to the given information:
The surface area of a cylinder is the sum of the lateral surface area (the curved surface) and the area of the two circular bases.
The formula for the lateral surface area of a cylinder is given:
Lateral Surface Area = 2πrh
where r is the radius of the cylinder and h is the height of the cylinder.
Plugging in the given values for the radius (r = 3 ft) and height (h = 2 ft), we can calculate the lateral surface area:
Lateral Surface Area = 2π * 3 * 2 = 12π ft²
The formula for the area of a circle (which represents the bases of the cylinder) is given:
Circle Area = πr²
Plugging in the given value for the radius (r = 3 ft), we can calculate the area of each circular base:
Circle Area = π * 3² = 9π ft²
Since there are two bases in a cylinder, we multiply this by 2 to account for both bases:
2 * Circle Area = 2 * 9π = 18π ft²
Now, we can add the lateral surface area and the area of the two bases to find the total surface area of the cylinder:
Total Surface Area = Lateral Surface Area + 2 * Circle Area
= 12π + 18π
= 30π ft²
As a decimal rounded to the nearest tenth, the surface area of the cylinder is approximately 94.2 ft²
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please help! and explain and show work of how you got the answer. I will mark you brainliest.
Answer: i dont know if this is right but i put a^4+b^4=c^4
Step-by-step explanation:
I have no idea tho
Show your work please
Answer:
Perimeter:
[tex]2( \frac{2}{3} + \frac{1}{2} ) = 2( \frac{4}{6} + \frac{3}{6} ) = 2( \frac{7}{6} ) = \frac{7}{3} = 2 \frac{1}{3} [/tex]
The perimeter is 2 1/3 inches.
Area:
[tex] \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} [/tex]
The area is 1/3 of a square inch.
lee had scored the following points in his first 8 games 12,14,14,15,8,10,3,15 enter the number of points lee needs to score in the nest game to increase his keam score to 13 points
Lee needs to score a total of 26 points in the next game to increase his mean score to 13 points.
Calculating the score in the next gameGiven that
Scores = 12,14,14,15,8,10,3,15
To increase his mean score to 13 points,
Lee needs to have a total of 13 x 9 = 117 points after the next game.
He has already scored a total of 12+14+14+15+8+10+3+15 = 91 points in the first 8 games.
Therefore, Lee needs to score a total of 117 - 91 = 26 points in the next game to increase his mean score to 13 points.
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PLEASE HELP DUE TODAY!!!!!!!
Consider the functions g(x) = 2x + 1 and h(x) = 2x + 2 for the domain 0 < x < 5
a. Without evaluating or graphing the functions, how do the ranges compare?
b. graph the 2 functions and describe each range over the given interval
Answer:
see the images and explanation
Step-by-step explanation:
for the graph:
the domain 0 < x < 5
the range for each functions:
g(x) = 2x + 1
g(x) = y , 1 < y < 11
h(x) = 2x + 2 , 2 < y < 12
triple the difference of c and b, then subtract the result from a
Therefore, if you triple the difference of c and b (which is 2), and then subtract the result from a (which is 10), you get the answer of 4.
The mathematical expression for "triple the difference of c and b, then subtract the result from a" is:
a - 3(c - b)
In this expression, "c" and "b" are variables representing numerical values, and "a" is another variable representing a numerical value.
To evaluate this expression, you would need to substitute specific numerical values for "a", "b", and "c". For example, if a = 10, b = 5, and c = 7, then the expression would become:
10 - 3(7 - 5) = 10 - 6 = 4
Therefore, if you triple the difference of c and b (which is 2), and then subtract the result from a (which is 10), you get the answer of 4.
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In morning Emily studied 40 minutes for a math exam. Later that evening Emily studied for x more minutes. Write an equation that represents the total number of minutes y emily studied for the math exam
Equation that represents the total number of minutes y Emily studied for the math exam is y = 40 +x.
We can represent the total number of minutes Emily studied for the math exam using the equation,
y = 40 + x
Here, 'y' represents the total number of minutes Emily studied for the math exam, '40' represents the number of minutes she studied in the morning, and 'x' represents the number of minutes she studied later that evening.
By adding the number of minutes studied in the morning to the number of minutes studied later that evening, we can calculate the total number of minutes Emily studied for the math exam.
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20. The wallpaper border that runs all the way around a room is 5f² +19f+ 11 long. Three sides of the room have the following lengths of border: 6f, 5f-7, 2f^2+2. What is the length of the fourth side of the room?
In linear equation, 3f² + 8f + 16 feet is the length of the fourth side of the room.
What is linear equation?
An algebraic equation with simply a constant and a first- order( direct) element, similar as y = mx b, where m is the pitch and b is the y- intercept, is known as a linear equation.
The below is sometimes appertained to as a" direct equation of two variables," where y and x are the variables. Equations whose variables have a power of one are called direct equations. One illustration with only one variable is where layoff b = 0, where a and b are real values and x is the variable.
= 5f² + 19f + 11 - 6f - 5f + 7 - 2f² - 2
= 3f² + 8f + 16 feet
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The expression that represent the length of the fourth side of the room is [tex]3f^2 + 8f + 16[/tex].
What is expression ?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. Unknown variables, integers, and arithmetic operators are the components of an algebraic expression. There are no symbols for equality or inequality in it.
Here Total length of the room = [tex]5f^2+19f+11[/tex]
Three sides length of the room = 6f , 5f-8 and [tex]2f^2+2[/tex]
Fourth side = Total length - sum of length of three sides
=> [tex]5f^2+19f+11-(6f+5f-7+2f^2+2)[/tex]
=> [tex]5f^2+ 19f + 11 - 6f - 5f + 7 - 2f^2 - 2[/tex]
=> [tex]3f^2 + 8f + 16[/tex] .
The expression that represent the length of the fourth side of the room is [tex]3f^2 + 8f + 16[/tex].
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patients scheduled to see their primary care physician at a particular hospital wait, on average, an additional eight minutes after their appointment is scheduled to start. assume the time that patients wait is exponentially distributed. what is the probability a randomly selected patient will see the doctor within eleven minutes of the scheduled time?
The probability is approximately 0.446 or 44.6%.
Waiting time is exponentially distributed, we can use the cumulative distribution function (CDF) of the exponential distribution to solve this problem.
Let X be the waiting time in minutes.
X is exponentially distributed with a mean of 8 minutes.
The rate parameter λ is equal to 1/8.
The CDF of the exponential distribution is given by:
[tex]F(x) = 1 - e^(-\lambda x)[/tex]
The probability that a patient will see the doctor within eleven minutes of the scheduled time.
The value of F(11) - F(0).
[tex]F(11) = 1 - e^(-\lambda \times 11) = 1 - e^(-11/8)[/tex] ≈ 0.554
[tex]F(0) = 1 - e^(-\lambda \times 0) = 1 - e^0 = 1[/tex]
Therefore, the probability that a randomly selected patient will see the doctor within eleven minutes of the scheduled time is:
F(11) - F(0) ≈ 0.554 - 1 ≈ 0.446
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Currently, Jana's grandmother is 5.25 times older than Jana.
Jana's mother is 2.5 times older than Jana. In 3 years, the sum
of all three of their ages will be 149 years. Write and solve an
equation to find Jana's current age. Then find her mother's
and grandmother's current ages.
Step-by-step explanation:
j = jana's age
m = mom's age = 2.5 j
g = grandma's age = 5.25 j summed, they equal 149 IN THREE YEARS
j+3 + 2.5j +3 + 5.25 j + 3 = 149
8.75 j + 9 = 149
8.75 j = 140
j = 16 y/o
mom = 2.5 j = 40 y/o
g-ma = 5.25 j = 84 y/o
Your friend incorrectly factors the expression below as 7x(5-2xy). Factor the expression below correctly. What error did your friend make?
35x - 14xy
Answer:
The answer is 7x(5-2y)
Step-by-step explanation:
35x-14xy
7x(5-2y)
the error made is the x in the bracket
for a normal distribution, a positive value of z indicates that group of answer choices all the observations must have had positive values. the area corresponding to the z is either positive or negative. the sample mean is smaller than the population mean. the sample mean is larger than the population mean.
For a normal distribution ,a positive value of z simply means that the observation or sample mean is above the population mean this implies
none of the options provided is completely accurate.
All the observations must have had positive values,
It is not necessarily true for a positive value of z.
The value of z indicates how many standard deviations away from the mean a particular observation or sample mean is.
A positive value of z simply means that the observation or sample mean is above the population mean.
The area corresponding to the z is either positive or negative,
It is also not accurate.
The area under the normal curve corresponds to probabilities, not positive or negative values.
The area to the right of the mean corresponds to positive z-values, and the area to the left of the mean corresponds to negative z-values.
The sample mean is smaller than the population mean,
It is a possibility when the z-value is negative, indicating that the sample mean is below the population mean.
The sample mean is larger than the population mean,
It is a possibility when the z-value is positive, indicating that the sample mean is above the population mean.
Therefore, the sample mean can be either larger or smaller than the population mean depending on the direction of the z-value.
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a prescription for ear drops specifies that the patient put 2 drops in each ear twice a day for 9 days. approx how many millilitres of medication would be needed for this prescription? ( dosage×frequency×day supply )
approximately 3.6 milliliters of medication would be needed for this prescription.
To calculate the total milliliters of medication needed for this prescription, we will use the formula dosage × frequency × day supply.
In this case, the dosage is 2 drops per ear, the frequency is twice a day, and the day supply is 9 days.
First, let's find out how many drops are needed per day:
2 drops/ear × 2 ears = 4 drops
Since the frequency is twice a day, multiply this number by 2:
4 drops × 2 = 8 drops per day
Now, multiply the number of drops per day by the day supply (9 days):
8 drops/day × 9 days = 72 drops
Typically, 1 drop is approximately 0.05 millilitres. To find out how many millilitres of medication are needed, multiply the total number of drops by the volume of one drop:
72 drops × 0.05 mL/drop = 3.6 mL
So, approximately 3.6 millilitres of medication would be needed for this prescription.
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Approximately 1.8 milliliters of medication would be needed for this prescription.
To calculate how many milliliters of medication would be needed for this prescription, we need to use the following formula:
dosage × frequency × day supply.
The dosage is 2 drops, the frequency is twice a day, and the day supply is 9 days.
First, we need to convert drops to milliliters.
One drop is equal to approximately 0.05 milliliters.
2 drops would be equal to 0.1 milliliters.
Next, we can plug in the values into the formula:
[tex]0.1 milliliters/drop \times 2 drops/twice a day \times 9 days = 1.8 milliliters. [/tex]
The patient will be using exactly 2 drops in each ear for each dose, and that there will be no waste or spillage of the medication.
To determine how many milliliters of medicine would be required for this prescription:
dose frequency day supply.
The recommended dosage is 2 drops, twice day, with a 9-day supply.
We must first convert droplets to milliliters.
A drop is around 0.05 milliliters in volume. As a result, 2 drops are equivalent to 0.1 milliliters.
The values may then be entered into the formula as follows:
0.1 milliliters each drop times twice daily for nine days equals 1.8 milliliters.
As a result, this prescription would require about 1.8 milliliters of medicine.
It's crucial to note that this estimate is predicated on the patient applying precisely 2 drops to each ear for each dose.
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A classmate of yours stated that a solid line is not a good representation of an arithmetic sequences. What logical assumption is your classmate using?
The classmate is not correct. A line is a good representation of an arithmetic sequence.
A line is a series of dots that represent each value of the sequence.
A line has the same slope as the common difference in the sequence.
An arithmetic sequence is a set of discrete values, whereas a line is a continuous set of values.
The logical assumption used is: An arithmetic sequence is a set of discrete values, whereas a line is a continuous set of values.
What is arithmetic sequence?An arithmetic sequence is a set of numbers where, with the exception of the first term, each term is obtained by adding a fixed constant to the term before it. Every pair of following terms in the sequence has the same fixed constant, which is known as the common difference. A1 stands for the first term in an arithmetic sequence, while an is used to represent the nth term.
A solid line symbolises continuous numbers, whereas the classmate's logical presumption is that an arithmetic series comprises of discrete values. This presumption is untrue, though, as a line can effectively represent an arithmetic series.
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Explain why the value of 0.6 is greater than the value of 0.39.
Answer:
B
Step-by-step explanation:
The value of 0.6 is greater than the value of 0.39 because 0.6 represents a larger proportion or fraction of a whole than 0.39 does.
To compare these two values, we can think of them as parts of a whole. For example, we can consider 0.6 as 60% of a whole and 0.39 as 39% of the same whole. When we compare 60% and 39%, it is clear that 60% is greater than 39%.
Another way to compare these two values is to convert them to fractions. 0.6 can be written as 6/10 or 3/5, while 0.39 can be written as 39/100.
When we compare 6/10 and 39/100, we can see that 6/10 is greater than 39/100 because 6/10 represents 6 parts out of 10, while 39/100 represents 39 parts out of 100.
Therefore, we can conclude that the value of 0.6 is greater than the value of 0.39 because it represents a larger proportion or fraction of a whole.
Jaxon made 5% of his free throws over the season. If he shot 220 free throws, how many did he make?
Answer:
Jaxon made 11 free throws over the season.
Step-by-step explanation:
If he made 5% of his free throws, we know that he will make 5% of the total number of free throws he took, which is 220:
We multiply 220 by 0.05, or 5% to find out how many free throws he made:
220*0.05 = 11
After that, we now know that Jaxon made 11 of his free throws out of 220 over the course of the season, or 5%.
An object moving at speed s for time t travels distance st. The Brown family is taking a road trip. This morning, they drove for 2. 5 hours at a speed of 65 miles per hour. How far did the family drive this morning?
Write your answer as a whole number or decimal
The total distance Brown family drove this morning at the speed of 65 miles per hour in 2.5 hours is equal to 162.5 miles.
Let 's' represents the speed of the moving object.
And 't' represents the time to cover distance.
The speed was 65 miles per hour.
And the time they drove was 2.5 hours.
The distance the Brown family drove this morning
= Multiplying their speed by the time they drove.
This implies,
distance = speed x time
Substitute the values we have,
The distance they drove this morning is equal to,
⇒ distance = 65 miles per hour x 2.5 hours
⇒ distance = 162.5 miles
Therefore, the distance drove by Brown family this morning is 162.5 miles at the speed of 65 miles per hour.
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Sarah took the advertising department from her company on a round trip to meet with a potential client. Including Sarah a total of 13 people took the trip. She was able to purchase coach tickets for $380
and first class tickets for $1200. She used her total budget for airfare for the trip, which was $9040. How many first class tickets did she buy? How many coach tickets did she buy?
number of first class tickets bought=
Answer: Sarah bought 5 first class tickets and (13-5) = 8 coach tickets.
Step-by-step explanation: The taken a toll of one to begin with lesson ticket is $1200 and the fetched of one coach ticket is $380.
So, the full taken a toll of first class tickets would be 1200x and the entire cost of coach tickets would be 380(13-x) = 4940 - 380x.
The overall fetched of the tickets is given as $9040, so we will set up the taking after condition:
1200x + 4940 - 380x = 9040
Streamlining and tackling for x, we get:
820x = 4100
x = 5
A city's yearly snowfall in inches over the last 5 years is recorded.
What is the mean of the snowfall in the last 5 years and how can it be interpreted?
Responses:
A mean of 25 shows that the data values are relatively close together. It represents that typically the city's snowfall has been about 25 inches yearly the last 5 years.
A mean of 25 shows that the data values are spread out. It represents that the city's snowfall has been exactly 25 inches each year for the last 5 years.
A mean of 27.2 shows that the data values are relatively close together. It represents that typically the city's snowfall has been about 27.2 inches yearly the last 5 years.
A mean of 27.2 shows that the data values are spread out. It represents that the city's snowfall has been exactly 27.2 inches each year for the last 5 years.
Year Snowfall (inches)
2015 25
2016 27
2017 25
2018 29
2019 30
Answer:
A mean of 27.2 shows that the data values are relatively close together. It represents that typically the city's snowfall has been about 27.2 inches yearly over the last 5 years.
Step-by-step explanation:
The mean, also known as the average, is the sum of all the data values divided by the number of data points. In this case, the mean of 27.2 inches indicates that, on average, the city has experienced a snowfall of approximately 27.2 inches per year over the last 5 years. The fact that the mean is close to the individual values of 25, 27, 25, 29, and 30 inches suggests that the data points are relatively close together and do not deviate significantly from the mean, indicating that the snowfall in the city has been relatively consistent over the past 5 years.
Question # 9
Multiple Choice
Estimate the difference.
10{1}{9} - 5{14}{15}
5
4{1}{2}
4
5{1}{2}
To estimate the difference 10 1/9 - 5 14/15, we can round both numbers to the nearest whole number and it gives (c) 4
What is the Estimated difference?From the question, we have the following parameters that can be used in our computation:
10 1/9 - 5 14/15
10 1/9 is between 10 and 11, but closer to 10 so we can round it to 10.Similarly, 5 14/15 is between 5 and 6, but closer to 6 so we can round it to 6.Using the above as a guide, we have the following:
The estimated difference is: 10 - 6
Evaluate
The estimated difference is: 4
Therefore, the Estimated difference is 5
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in conditional statements, the part of the statement following ‘if’ is called ___antecedent or consequent
The part of the statement following "if" is called the antecedent, and the part of the statement following "then" is called the consequent in conditional statements.
The if statement evaluates the test expression inside the parenthesis ().
If the test expression is evaluated to true, statements inside the body of if are executed.
If the test expression is evaluated to false, statements inside the body of if are not executed.
The part of the statement following "if" is called the antecedent, and the part of the statement following "then" is called the consequent in conditional statements.
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In conditional statements, the part of the statement following 'if' is called the antecedent.
The antecedent is the condition that needs to be true for the consequent to occur.
The consequent is the part of the statement that follows 'then.'
An antecedent is a noun or pronoun that denotes a specific being, place, object, or clause.
It's also referred to as a referent. Without an antecedent, a sentence may be insufficient or nonsensical since it is
required to establish what or to whom a pronoun in a sentence is referring.
In summary, a conditional statement is structured as "if (antecedent) then (consequent)."
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three six-sided, fair dice are rolled. what is the probability that none of the outcomes is an even number
the probability that none of the outcomes is an even number is 1/8 or approximately 0.125
The probability of rolling an even number on a fair six-sided die is 3/6 or 1/2, and the probability of rolling an odd number is also 1/2.
To find the probability that none of the outcomes is an even number, we need to calculate the probability of rolling an odd number on all three dice. Since the dice are rolled independently, we can multiply the probabilities together to get the overall probability:
P(rolling odd on one die) = [tex]1/2[/tex]
P(rolling odd on two dice) = [tex](1/2)^2 = 1/4[/tex]
P(rolling odd on three dice) =[tex](1/2)^3 = 1/8[/tex]
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help Here is your graph of the points on the previous screen.
Connect the points in order to create polygon `ABCDEF`.
1.2 Enter the length of the segment betwee
The length of the given line AB is 6 units and the polygon ABCDEF has been created.
What is a graph?A graph is a mathematical structure made up of a collection of points called VERTICES and a set of lines connecting some pair of VERTICES that may or may not be empty.
There is a chance that the edges will be directed, or orientated.
If the lines are directed or undirected, respectively, they are referred to as ARCS or EDGES.
Make a sequence of bars on graph paper as an example of a graph.
So, in the given situation the polygon ABCDEF has been created:
(Refer to the graph attached below)
Now, the length of the side AB:
Count the units as follows which comes to 6 units.
Therefore, the length of the given line AB is 6 units and the polygon ABCDEF has been created.
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Correct question:
Help Here is your graph of the points on the previous screen.
Connect the points in order to create polygon `ABCDEF`.
1.2 Enter the length of the segment between A and B.
Please answer this geometry question
The value of cos ( 90 - x ), given the side lengths of the right triangle would be B. 3 / 5.
How to find the value of the cosine ?In a right triangle, the sum of the two acute angles is always 90 degrees. If we have a right triangle with side lengths 3, 4, and 5, then it's a 3-4-5 Pythagorean triple, which means the angles are also in a specific ratio.
Now, we want to find cos(90 - x). Since the sum of the two acute angles in a right triangle is 90 degrees, the other angle in the triangle is (90 - x). We know that sin x = opposite / hypotenuse, so in our triangle, sin(90 - x) = 3/5.
The sine and cosine of complementary angles are related by the co-function identity:
cos(90 - x) = sin x
So, cos(90 - x) = sin(90 - x) = 3/5.
The answer is 3/5.
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What are the domain and range of this exponential function?y=2x–9
The value of both domain and range of this exponential function is (−∞,∞).
y = 2x-9
The domain of the function can be defined as all real numbers except the ones where the expression is undefined. In the case of 2x-9, there is no real number for which this expression is undefined. Therefore, a domain of this exponential function is (−∞,∞).
The range of the function is defined as the set of all valid y values. In this case, all real numbers are valid values of y. Therefore, the range of this exponential function is (−∞,∞).
Therefore, domain of y = 2x-9 is (−∞,∞) and range is also (−∞,∞).
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I need some help bro
Answer:
first answer is correct
y = -1/2 x^2 + x + 1
Step-by-step explanation:
based on the graph, you can find two points
(2,1) and (0,1)
I input (2, 1) to each equation to check
y = -1/2 x^2 + x + 1 is correct
Question #5 I really don’t get it I would appreciate an answer it’s been an hour and no response
The value of x is 11 and the measure of the supplementary angle m∠N is equal to 143°. That is:
a. equation:37° + 3x = 180°
b. x = 11, m∠M = 37°, m∠N = 143°
What are supplementary anglesSupplementary angles are a pair of angles that add up to 180 degrees. In other words, if you have two angles that are adjacent (share a common vertex and side) and their measures add up to 180 degrees, then they are supplementary angles.
Given that m∠M and m∠N are supplementary then:
37° + 3x = 180°
3x = 180° - 37° {collect like terms}
3x = 143°
x = 143/3 {divide through by 3}
x = 11
m∠N = 3(11) = 143°.
Therefore, the value of x is 11 and the measure of the supplementary angle m∠N is equal to 143°.
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What is the value of g(x)=x^2+4x
The value of g(x) = x^2 + 4x depends on the value of x.
If, for example, x = 2, then:
g(2) = 2^2 + 4(2) = 4 + 8 = 12
If x = -3, then:
g(-3) = (-3)^2 + 4(-3) = 9 - 12 = -3
In general, we can evaluate g(x) for any value of x by substituting that value into the expression x^2 + 4x.
a frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of $3$, or to the closest point with a greater integer coordinate that is a multiple of $13.$ a move sequence is a sequence of coordinates which correspond to valid moves, beginning with $0,$ and ending with $39.$ for example, $0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39$ is a move sequence. how many move sequences are possible for the frog?
There are 16 possible move sequences for the frog.
The possible moves that the frog can make.
The frog can move to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13.
Let's consider the first type of move, where the frog moves to the closest point with a greater integer coordinate that is a multiple of 3.
If the frog is at the point [tex]$n$[/tex], then the closest point with a greater integer coordinate that is a multiple of 3 is either [tex]$n+3$ or $n+6$[/tex].
Similarly, if the frog is at the point [tex]$n$[/tex] and wants to move to a multiple of 13, then the closest point with a greater integer coordinate that is a multiple of 13 is either[tex]$n+13$[/tex] or [tex]$n+26$[/tex].
Constructing move sequences.
The sequence starts with 0 and ends with 39.
The frog has two choices for its first move: either move to 3 or move to 13.
Let's assume that the frog moves to 3.
From there, the frog has two choices again:
move to 6 or move to 13. If the frog moves to 6, then it has two choices again: move to 9 or move to 13.
If it moves to 13, then it has two choices: move to 26 or move to 16.
We can continue this process until we reach 39.
We can create a tree diagram to help us keep track of the possible move sequences:
0
|
3 or 13
/ \
6 or 13 13 or 26
/ \ / \
9 13 16 26
/ \ / \ / \ / \
... ... ... ... ... ... ... ...
Each branch represents a different move sequence.
We can see that there are 16 possible move sequences in total.
We can also see that some sequences are longer than others.
The longest sequence has 7 moves, while the shortest sequence has only 2 moves.
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