By solving a system of equations, we will see that the function is continuous for all values of x if:
a = 2.5b = -0.5.How to make the piecewise-function continuous for every x?Here we have the piecewise function:
f(x) = -7 if x ≤ -3
f(x) = a*x- b if -3 < x < 1
f(x) = 3 if 1 ≤ x
The piecewise function will be continuous only if the values at the limits where we have jumps (at x = -3 and x = 1) are the same in both pieces of the function.
This means that we must have:
f(-3) = -7 = a*(-3) - b
f(1) = 3 = a*1 - b
Then we have a system of equations:
-7 = -3a - b
3 = a - b
To solve this, first, we isolate one of the variables in one of the equations, I will isolate b in the second one to get:
b = a - 3
Now we can replace that in the other equation to get:
-7 = -3a - (a - 3)
Now we can solve this for a.
-7 = -4a + 3
-7 -3 = -4a
-10/-4 = a = 2.5
Then the value of a must be 2.5, to get the value of b we use:
b = a - 3 = 2.5 - 3 = -0.5
So the solution is a = 2.5, b = -0.5.
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Which function has the widest graph?
A. y=2x^2 B. y=x^2 C. y=0.5x^2 D.y=-x^2
Do you have to graph each one to find the answer? ...?
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?
Slope is - 3 And (0,-4) is on the line
Which of the following options is an equivalent function to f(x) = 3(2)3x?
f(x) = 3(8)x
f(x) = 24x
f(x) = 27(8)x
f(x) = 3(8x)
The correct option is A: [tex]\( \boxed{f(x) = 3(8)^x} \).[/tex]
To determine which option is equivalent to the given function f(x) = [tex]3(2)^{3x} \),[/tex] we will simplify the given function and compare it to each option.
The given function is:
[tex]\[ f(x) = 3(2)^{3x} \][/tex]
We can rewrite the exponent part using properties of exponents:
[tex]\[ (2)^{3x} = (2^3)^x = 8^x \][/tex]
So the function becomes:
[tex]\[ f(x) = 3 \cdot 8^x \][/tex]
Now, let's compare this to each of the given options:
Option A: [tex]\( f(x) = 3(8)^x \)[/tex]
This is exactly the same as our simplified function. Therefore, Option A is equivalent to [tex]\( f(x) = 3(2)^{3x} \).[/tex]
Option B: [tex]\( f(x) = 24^x \)[/tex]
This is not equivalent because it implies a different base raised to the power of x.
Option C:[tex]\( f(x) = 3(8x) \)[/tex]
This is not equivalent because it multiplies by 27 instead of 3.
Option D: [tex]\( f(x) = 3(8x) \)[/tex]
This is not equivalent because it implies multiplying 8 by x, not raising 8 to the power of x.
Therefore, the correct answer is:
[tex]\( \boxed{f(x) = 3(8)^x} \).[/tex]
The complete question is here:
Which of the following options is an equivalent function to f(x) = 3(2)3x?
A. f(x) = 3(8)x
B. f(x) = 24x
C. f(x) = 27(8)x
D. f(x) = 3(8x)
Solve the differential equation using the method of undetermined coefficients
a) 4y"- 4y'-3y= cos2x
-1.4n + 2.1 = 6.58
what is n
El area del cuadrado mide 25 pies cuadrados cuál es la longitud de un lado del cuadrado?
. To change the shape of a function, you need to _____ or compress it.
A. compress
B. stretch
C. shift
D. flip
Answer:
B. Strech
Step-by-step explanation:
Stretching can change the shape of something such as a function.
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
Which graph represents a reflection of f(x) = 2(0.4)x across the y-axis?
Answer: The correct option is A.
Explanation:
The given function is,
[tex]f(x)=2(0.4)^x[/tex]
To find the graph of this function after the reflecting across y-axis, first we have to find the graph of the equation.
The value of the function is 2 when x=0, so, the graph of given equation intersect the y-axis at 2.
In the equation [tex](0.4)^x[/tex]. Since [tex]0<0.4<1[/tex], so the given function is decreasing function.
[tex]f(x)\rightarrow 0 \text{ as }\rightarrow \infty[/tex]
[tex]f(x)\rightarrow \infty \text{ as }\rightarrow -\infty[/tex]
The value of f(x) is always positive, so the graph of f(x) is always above the x-axis. Thus, the graph must be above the x-axis after reflection across y-axis.
So, the option (2) and (4) and incorrect.
When we reflect the graph across the y-axis then,
[tex]f(x)\rightarrow \infty \text{ as }\rightarrow \infty[/tex]
[tex]f(x)\rightarrow 0 \text{ as }\rightarrow -\infty[/tex]
It means when x approaches to large negative number the f(x) approaches to 0 and when x approaches to large positive number the f(x) approaches to infinite.
Therefore, the correct option is show in first graph.
Answer:
a
Step-by-step explanation:
Solve the system of equations and choose the correct ordered pair.
3x + 2y = 12
6x + 3y = 21
A. (4, 3)
B. (4, 0)
C. (2, 3)
D. (3, 2)
Answer:
C.
Step-by-step explanation:
We are going to solve it with substitution, that is, we are going to clear one variable and then replace it in the other equation. Then,
[tex]3x + 2y = 12[/tex]
[tex]3x = 12-2y[/tex]
[tex]x = \frac{12-2y}{3}[/tex]
[tex]x = \frac{12}{3}-\frac{2y}{3}[/tex]
[tex]x = 4-\frac{2y}{3}.[/tex]
Now, we replace that x value in the equation 6x + 3y = 21:
[tex]6(4-\frac{2y}{3}) + 3y = 21[/tex]
[tex]24-\frac{12y}{3} + 3y = 21[/tex]
[tex]24-4y + 3y = 21[/tex]
[tex]24-y = 21[/tex]
[tex]-y = 21-24[/tex]
[tex]-y = -3[/tex]
[tex]y = 3.[/tex]
Finally, we replace the y value found to find x.
[tex]x = 4-\frac{2*3}{3}[/tex]
[tex]x = 4-2[/tex]
[tex]x = 2.[/tex]
So, the solution is (x,y)=(2,3). Then, the answer is C.
help please?
solve for Y
a(y+c)=b(y-c)
solve for W
A=lw+wh+lh
...?
The following data values represent a population. What is the variance of the values?
8, 10, 14, 12
A.22
B.11
C.5
D.10
Answer:
It’s actually 5
Step-by-step explanation:
Ape x
What is the first step to writing an equation of a line when given two points?
a factory makes 1200 pounds of potato chips in four hours at this rate how many pounds of potato chips will a factory make in 10 hours
A factory producing 1200 pounds of chips in four hours will yield 3000 pounds when operating for 10 hours, assuming a consistent production rate.
Explanation:The given problem pertains to rate, which allows the determination of how much of something can be done in a specific amount of time. In this case, the rate of potato chip production is 1200 pounds every four hours. To find out how many pounds of potato chips can be produced in 10 hours, we first need to determine the hourly rate of production.
Firstly, we determine the hourly production rate of the factory by dividing the total produce by the time spent. That is 1200 pounds / 4 hours = 300 pounds/hour.
At this hourly rate, the amount of potato chips that can be produced in 10 hours can be calculated by multiplying the hourly rate by the number of hours, which is 300 pounds/hour * 10 hours = 3000 pounds.
So, a factory that produces 1200 pounds of potato chips in four hours will make 3000 pounds of potato chips in 10 hours.
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The Sugar Sweet Company is going to transport its sugar to market. It will cost
$4500 to rent trucks, and it will cost an additional $175 for each ton of sugar transported. Let C represent the total cost (in dollars), and let S represent the amount of sugar (in tons) transported. Write an equation relating C to S. Then use this equation to find the total cost to transport 19 tons of sugar.
4 questions that I don't understand how to do. Help please?
**!!!STUCK ON 2 QUESTIONS!!!!
The time is takes to mow the lawn at a large park m(x) varies inversely with the numbers of workers assigned to the job x. It takes 90 minutes to complete the job when 3 workers are assigned to it.
which equation can be used to find the time to complete the job when x workers are assigned to it?
A.) m(x)= 270/x
B.) m(x) = 270x
C.) m(x) = 30/x
D.) m(x) = 30x
2. Suppose that H(x) varies inversely with x and H(x)=50 when x =0.25
What is H(x) when x =2?
A.) 0.5
B.) 6.25
C.) 12.5
D.) 24
Can you also provide how you got the answer as well :)
Answer:
1. Which equation can be used to find the time to complete the job when x workers are assigned to it?
A.) m(x) = 270/x
2. Suppose that H(x) varies inversely with x and H(x)=50 when x =0.25
What is H(x) when x =2?
B.) 6.25
Step-by-step explanation:
1. We know when x=3 (workers), m(x)= 90 min (time is takes to mow the lawn at a large park) and we know m(x) varies inversely with x, this means when m(x) increases so x decreases. This is obtained with a division over x.
Now, we can replace to know what is the correct answer:
[tex]m(x)=\frac{y}{x} (1) \\90=\frac{y}{3} \\90*3=y[/tex]
[tex]y=270[/tex]
The answer is A.
[tex]m(x)=\frac{270}{x}[/tex]
We can confirm if we replace x=3 in the equation before. Our answer should be 90.
[tex]m(x)=\frac{270}{3}=90[/tex]
2. Suppose that H(x) varies inversely with x and H(x)=50 when x =0.25
We know H(x) varies inversely with x, this means when H(x) increases so x decreases. This is obtained with a division over x. Now, we need to find the constant in the numerator.
Now, we can replace in the eq (1) to know the constant in the numerator:
[tex]H(x)=\frac{y}{x}\\50=\frac{y}{0.25} \\50*0.25=y[/tex]
[tex]y=12.5[/tex]
Now, we can replace in the eq (1) to find H(x) when x=2
[tex]m(x)=\frac{y}{x}[/tex]
[tex]m(x)=\frac{12.5}{2}[/tex]
[tex]m(x)=6.25[/tex]
The answer is B.) 6.25
The formula for an object's final velocity is f=i−gt f=i−gt . Solve for t.
answer: (f-i)/-g=t
I hope it helps~
What is the midpoint of 0.7 0.8?
Which fraction is NOT equivalent to 4/12?
A. 1/3 B. 12/30 C. 8/24 D. 16/48
keisha is reading a 325 page book at a rate of 25 pages per day. use a point slipe equation to determine whether she will finish reading the book in 10 days.
Using the point slope equation, it is determined that Keisha will not finish reading the 325 pages book in 10 days if she is reading at a rate of 25 pages per day.
Explanation:The subject of this question is Mathematics and it involves solving a problem using a point slope equation.
Let's denote the total number of pages in the book as 'y' and the number of pages that Keisha can read per day as 'm'. We know that y=325 and m=25. The point slope equation in this case would be y=mx, where 'x' represents the number of days. In this case, we are trying to find out whether x=10.
Substituting the known values into the equation we get: 325=25*10. This simplifies to 325=250.
Since 325 does not equal 250, we can conclude that Keisha will not complete reading the book in 10 days.
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Two particles are fixed to an x axis: particle 1 of charge -1.00 x 10-7 C is at the origin and particle 2 of charge +1.00 x 10-7 C is at x = 17.1 cm. Midway between the particles, what is the magnitude of the net electric field?
...?
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.
I'm in high school and need help with this problem. 2x +y = -13 x= 3x+y
Your current account balance is $215. You have $322 of expenses each month. Your income is $444 per month. How long will it take you to accumulate a balance of $1000 in your account.
It take 6.434 month or 6 month and 13 days you to accumulate a balance of $1000 in your account.
What is word problem in mathematics?Word problem is a mathematical problem totally described in language that is frequently used as an educational tool.
Your current account balance is $215.
Your per month income is $444.
Your per month expense is $322.
Hence, your per month savings is = $444 - $322 = $122.
Let, it take x month you to accumulate a balance of $1000 in your account.
Then mathematically,
215 + 122 x = 1000
Rearrange the equation can be written that:
122 x = 1000 -215
After the subtraction, the result will be:
122 x = 785
Rearrange the equation can be written that:
x = 785/122
After the division, the result will be:
x = 6.434 = 6 month and 13 days.
Hence, it take 6.434 month or 6 month and 13 days you to accumulate a balance of $1000 in your account.
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how to estimate 184÷9 in multiplication
Mark all of the statements that are true.
(will attach graph shortly)
a. The range for this function is the set {3}.
b. This graph is not a function because the value of y is the same for every value of x.
c. The domain for this function is all real numbers.
d. The domain for this function is the set {3}.
e. All real numbers are in the range of this function.
***CHOOSE ALL THAT APPLY!! And please explain!! :)
which correctly describes the root of the following cubic equation? x^3-3x^2+4x-12=0
Answer:
The equation has one real root and two complex roots.
Step-by-step explanation:
The given equation is
[tex]x^3-3x^2+4x-12=0[/tex]
The above equation is true form x=3, therefore (x-3) is a factor of above equation.
Use long division or synthetic division method to divide the equation by (x-3).
[tex](x-3)(x^2+4)=0[/tex]
Equate each factor equal to zero.
[tex]x-3=0[/tex]
[tex]x=3[/tex]
Therefore 3 is a real root of the equation.
[tex]x^2+4=0[/tex]
[tex]x^2=-4[/tex]
[tex]x=\sqrt{-4}[/tex]
[tex]x=\pm 2i[/tex]
2i and -2i are complex roots of the equation.
Therefore the equation has one real root and two complex roots.
Which number has a cube root between 7 and 8?
A.57
B.153
C.244
D.499