Answers
the domain is the interval or set of all valid values for x (the input variable).
the range is the interval or set of all valid values for y or f(x) (the function result variable).
the functional result values for the first piece :
x - 4 for 0 <= x < 2
for x = 0 we get y = -4
for x = 2 we get y = -2
the range for this piece is [‐4, -2).
we use the round bracket to indicate -2 itself is not included, as we have x < 2 (and not x <= 2).
the functional result values for the second piece :
x² - 3x + 4 for 2 <= x < 4
for x = 2 we get 4-6+4 = 2
for x = 4 we get 16-12+4 = 8
the range for this piece is [2, 8)
again, because of x < 4 we use the ")" for 8 (not included).
the functional result values for the third piece :
5 for 4 <= x < 7
the range is simply [5) = [5] because the set contains only one element.
and 5 is already contained in the range of the second piece.
so, the range is just the combination of the ranges of the first 2 pieces :
[-4, -2) U [2, 8)
Related Questions
help me with my work please
Answers
In the first one, the word increase means you will be add something
+42000
In the second one, the word withdrew means you will subtract something
-40
At a sale a sofa is being sold for 67% of the regular price. The sale price is $469. What is the regular price?
Answers
We have the following information:
• At a sale, a sofa is being sold for ,67% of the regular price
,• The sale price is $469
And we need to determine the regular price.
To find it, we can proceed as follows:
1. Let x be the regular price. Then we have:
[tex]\begin{gathered} 67\%=\frac{67}{100} \\ \\ 67\%(x)=\frac{67}{100}x \\ \\ \text{ Then we know:} \\ \\ \frac{67}{100}x=\$469 \end{gathered}[/tex]2. Now, we have to solve for x as follows:
[tex]\begin{gathered} \text{ Multiply both sides by }\frac{100}{67}: \\ \\ \frac{67}{100}*\frac{100}{67}x=\frac{100}{67}*\$469 \\ \\ x=\frac{100*\$469}{67}=\$700 \\ \\ x=\$700 \end{gathered}[/tex]Therefore, in summary, the regular price is $700.
Name: Date: Writing Linear Equations from Word Problems (Page 2) 5) You are planning a class pool party and have reserved the pool for $625. It will cost an additional $11 per person for food. Write a linear equation to represent the total cost of the pool party based on the number of people that will attend. a) If 26 people end up coming to the pool party how much will it cost you?
Answers
Consider that the reservation costs $625. Moreover, it is necessary to pay additional $11 per person for food. If x is the number of people, it means that it is necessary to pay 11x dollars for x people.
THen, the total cost is $625 plus 11x dollars. If y is the cost of the class pool party, you have the following algebraic equation which expresses the given situation:
y = 11x + 625
If there are 26 people in the party, that is, if x = 26, the cost is:
y = 11(26) + 625
y = 286 + 625
y = 911
Hence, the cost is $911
Domain and range of the quadratic function F(x)=-4(x+6)^2-9
Answers
Answer:
Domain: (-∞, ∞)
Range: (-∞, -9]
Step-by-step explanation:
how many irrational numbers are there between 1 and 6? is it infinite?
Answers
infinite numbers
Explanation
An Irrational Number is a real number that cannot be written as a simple fraction,for example Pi()
[tex]\pi=3.141592654[/tex]Step 1
between 0 an 6 we have 6 integers numbers :(1,2,3,4,5,6)
Step 2
Now check this
a number for example
[tex]\begin{gathered} 3.14 \\ is\text{ different to } \\ 3.145 \\ and\text{ it is diferrent to} \\ 3.1458 \end{gathered}[/tex]so, the answer is infinite numbers
Abigail and her friend Jaya are going to a carnival that has games and rides. Abigail played 7 games and went on 2 rides and spent a total of $24.75. Jaya played 3 games and went on 6 rides and spent a total of $33.75. Determine the cost of each game and the cost of each ride.
Answers
The cost of each game is $2.25.
The cost of each ride is $4.50.
What is the cost of each game and each ride?Here are the system of linear equations that can be derived from the question:
7g + 2r = 24.75 equation 1
3g + 6r = 33.75 equation 2
Where:
g = price of each game r = price of each rideIn order to determine the value of g, multiply equation 1 by 3:
21g + 6r = 74.25 equation 3
Subtract equation 2 from equation 3:
40.50 = 18g
g = 40.50 / 18
g = 2.25
In order to determine the value of r, substitute for g in equation 1 :
7(2.25) + 2r = 24.75
15.75 + 2r = 24.75
2r = 24.75 - 15.75
2r = 9
r = 9 /2
r = 4.50
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−|a+b|/2−c when a=1 2/3 , b=−1 , and c=−3
ENTER YOUR ANSWER AS A SIMPLIFIED FRACTION IN THE BOX.
Answers
The value of the expression −|a + b|/2 − c when a =1 2/3 , b=−1 , and c=−3 is 8/3
How to evaluate the expression?From the question, the expression is given as
−|a + b|/2 − c
Also, we have the values of the variables to be
a =1 2/3 , b=−1 , and c=−3
Rewrite a as
a = 5/3
So, we substitute a = 5/3 b=−1 , and c=−3 in −|a + b|/2 − c
This gives
−|a + b|/2 − c = −|5/3 - 1|/2 + 3
Evaluate the difference in the expression
−|a + b|/2 − c = −|2/3|/2 + 3
Divide
−|a + b|/2 − c = −|1/3| + 3
Remove the absolute bracket and solve
−|a + b|/2 − c = 8/3
Hence, the solution is −|a + b|/2 − c = 8/3
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The required simplified value of the given expression is 8/3.
As per the given data, an expression −|a+b|/2−c is given when a=1 2/3, b=−1, and c=−3 the value of the expression is to be determined.
The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
Let the solution be x,
x = −|a+b|/2−c
Substitute value in the above equation,
x = - |1 + 2 / 3 - 1|/2 - (-3)
x = -1/3 + 3
x = -1+9 / 3
x = 8/3
Thus, the required simplified value of the given expression is 8/3.
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Find the equation for the line that passes through the point (−2,5), and that is perpendicular to the line with the equation x=−4.
Answers
The equation of the line perpendicular to x = -4 that passes through the point (-2,5) is y = 5 .
In the question ,
it is given that
the required line is perpendicular to x = -4
the slope of x = -4
x + 4 = 0
0.y = x + 4
slope = 1/0
so the slope of the perpendicular line [tex]=[/tex] 0 .
the equation of the perpendicular line passing through (-2,5) and slope as 0 is
(y - 5) [tex]=[/tex] 0*(x + 2)
y -5 = 0
y = 5
Therefore , The equation of the line perpendicular to x = -4 that passes through the point (-2,5) is y =5 .
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An initial investment of $200 is appreciated for 20 years in an account that earns 6% interest, compounded continuously. Find the amount of money in the account at the end of the period.
Answers
To calculate the final amount of money at the end of the period, considering that the interest is compounded continuously you have to use the following formula:
[tex]A=P\cdot e^{rt}[/tex]Where
A is the accrued amount at the end of the given time
P is the principal amount
r is the annual nomial interes expressed as a decimal value
t is the time period in years
For this investment, the initial value is P= $200
The interest rate is 6%, divide it by 6 to express it as a decimal value
[tex]\begin{gathered} r=\frac{6}{100} \\ r=0.06 \end{gathered}[/tex]The time is t= 20 years
[tex]\begin{gathered} A=200\cdot e^{0.06\cdot20} \\ A=200\cdot e^{\frac{6}{5}} \\ A=664.02 \end{gathered}[/tex]After 20 years, the amount of money in the account will be $664.02
A word problem made out of the equation 1/4+3-7/8x<-2
Answers
The word problem of the expression is "Seven eighths of a number less three and a quarter is less than 2"
How to determine the word problem?The inequality expression is given as
1/4 + 3 - 7/8x < 2
From the above expression, we can see that:
The right-hand side of the inequality is 2 and the inequality symbol is less than
This means that the word problem must end in "less than 2"
Next, we analyse the left-hand side
1/4 + 3 - 7/8x
A statement that represents the above is
Seven eighths of a number less three and a quarter is less than 2
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Which of the following identities is used to expand the polynomial (3x - 4y)2?
Answers
Recall that to expand a binomial we can use the following formula:
[tex](a+b)^2=a^2+2ab+b^2.[/tex]The above is known as the square of the binomial.
Answer:
Square of binomial.
Given g(x) = -2√x², find g(8 + x).
Answers
Answer:
g(8 + x) = - 2(8 + x)
Step-by-step explanation:
substitute x = 8 + x into g(x)
g(8 + x) = - 2[tex]\sqrt{(8+x)^2}[/tex] = - 2(8 + x)
Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval 3≤x≤6
Answers
STEP - BY - STEP EXPLANATION
What to find?
Rate of change of the given function.
Given:
Step 1
State the formula for rate of change.
[tex]Rate\text{ of change=}\frac{f(b)-f(a)}{b-a}[/tex]Step 2
Choose any two point within the given interval.
(3, 59) and (6, 44)
⇒a=3 f(a) =59
b= 6 f(b)=44
Step 3
Substitute the values into the formula and simplify.
[tex]Rate\text{ of change=}\frac{44-59}{6-3}[/tex][tex]=\frac{-15}{3}[/tex][tex]=-5[/tex]ANSWER
Rate of change = -5
The right arrow symbol used to show the transition from a point to its image after a transformation is not contained within the Equation Editor. If such a symbol is needed, type "RightArrow." For example: P(0, 0) RightArrow P′(1, 2).
Triangle ABC has coordinates
A
(
1
,
4
)
;
B
(
3
,
−
2
)
;
and
C
(
4
,
2
)
.
Find the coordinates of the image
A
'
B
'
C
'
after a reflection over the x-axis.
Answers
Answer:
Step-by-step explanation:
Given:
Vertices of triangle ABC are A (1,4), B(3,−2) and C(4,2).
Triangle ABC reflected over the x-axis to get the triangle A'B'C'.
To find:
The coordinates of the image A'B'C'.
Solution:
If a figure reflected over the x-axis, then rule of transformation is
Now, using this rule, we get
Therefore, the coordinates of the image A'B'C' after a reflection over the x-axis are A'(1,-4), B'(3,2) and C'(4,-2).
how do I find the first five terms of the geometric sequence?
Answers
To find the next term of a geometric sequence, the previous term is multiplied by the common ratio r.
[tex]\begin{gathered} r=\frac{1}{2} \\ a_{1=}20 \\ a_2=20\times\frac{1}{2}=10 \\ a_3=10\times\frac{1}{2}=5 \\ a_4=5\times\frac{1}{2}=\frac{5}{2} \\ a_5=\frac{5}{2}\times\frac{1}{2}=\frac{5}{4} \end{gathered}[/tex]A visitor in a Las Vegas casino lost $200, won $100, and then lost $50. What was the change in the gambler's net worth (how much did the gambler win or lose overall)?
Answers
The gambler in a Las Vegas casino lost $150.
According to the question,
We have the following details:
The visitor lost $200 and $50 in a Las Vegas casino.
The visitor won $100.
So, in total, the visitor lost $250 overall.
Now, compare the amount the gambler lost and the gambler won in the casino. So, we notice that the gambler lost more amount.
Now, we will find the total amount lost:
Gambler's net worth = The amount lost - the amount won
Gambler's net worth = $(250-100)
Gambler's net worth = $150
Hence, the gambler lost $150 overall in the casino.
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A gardener makes a new circular flower bed. The bed is ten feet in diameter. Calculate the circumference and the area of the circular flower bed.
Answers
Answer:
C = 31.4159
A = 78.5398
Step-by-step explanation:
diameter = 10
radius = 5
plug into formula: C = 2π5
A = π * 5²
Trisha is using the recipe show to make a fruit salad. She wants to use 20 diced strawberries in her fruit salad. How many bananas, apples, and pears show Trisha use in her fruit salad? Fruit Salad Recipe 4 bananas 3 apples 6 pears 10 strawberries bananas apples pears
Answers
The original recipe has half of the number of strawberries She wants to use in her salad in this case we will need double the fruits of the recipe given
Bananas
4x2=8 bananas
Apples
3x2=6 apples
Pears
6x2=12 pears
(a) Write the number 302.658 correct to 2 decimal places, (b) Write the number 302.658 correct to 2 significant figures.
Answers
The given number is,
302.658
to corrct to 2 decimal places,
302.66.
Please help meeeeeeeeeeeeeeee
Answers
Answer:
-2 1/4 or 2.25
Step-by-step explanation:
i hope this helped and ita also self explanitory
Harry HAS 1 1/2 KG whole-wheat flour. He uses 3/4 of the flour to bake bread. How much flour did he use?
Answers
Given the following parameter
Mass of flour wheat flour = 1 1/2 kg
If Harry used 3/4 of these flour to bake bread, the amount of flour used is expressed as:
[tex]A=\frac{3}{4}\text{ of 1}\frac{1}{2}[/tex]Convert the mixed fraction into an improper fraction to have:
[tex]\begin{gathered} A=\frac{3}{4}\times\frac{3}{2}kg \\ A=\frac{9}{8}kg \\ A=1\frac{1}{8}kg \end{gathered}[/tex]This shows that Harry used 1 1/8kg of the wheat flour to bake the bread
The graph of F(x), shown below, has the same shape as the graph ofG(x) = x - x? Which of the following is the equation of F(x)?F(x) = ?
Answers
The graph of G(x) is symmetric with respect to the y-axis. It also passes through the origin.
Since the graph of F(x) moved 4 units upward, we must add 4 to the right of the equation. Thus, the equation of F(x) is as follows.
[tex]F(x)=x^4-x^2+4[/tex]9. Rosie's Bakery just purchased an oven for $1,970. The owner expects the oven to last for 10years with a constant depreciation each year. It can then be sold as scrap for an estimatedsalvage value of $270 after 10 years. (20 points)a) Find a linear equation modeling the value of the oven, y, after x years of use.b) Find the value of the oven after 2.5 years.c) Find the y-intercept. Explain the meaning of the y-intercept in the context of this problem.d) Graph the equation of the line. Be sure to label the axes.
Answers
a)
The oven devaluated from $1970 to $270 in 10 years.
Since each year it looses the same value, divide the change in the price over the time interval to find the rate of change of the value with respect to time.
To find the change in price, substract the initial price from the final price:
[tex]270-1970=-1700[/tex]The change in price was -$1700.
Divide -1700 over 10 to find the change in the price per year:
[tex]-\frac{1700}{10}=-170[/tex]The initial value of the oven was $1970, and each year it looses a value of $170.
Then, after x years, the value will be equal to 1970-170x.
Then, the linear equation that models the value of the oven, y, after x years of use, is:
[tex]y=-170x+1970[/tex]b)
To find the value of the oven after 2.5 years, substitute x=2.5:
[tex]\begin{gathered} y_{2.5}=-170(2.5)+1970 \\ =-425+1970 \\ =1545 \end{gathered}[/tex]Then, the value of the oven after 2.5 years is $1545.
c)
To find the y-intercept, substitute x=0:
[tex]\begin{gathered} y_0=-170(0)+1970 \\ =1970 \end{gathered}[/tex]The y-intercept is the initial value of the oven when 0 years have passed.
d)
Is-7+9 = -9 + 7 true, false, or open?
Answers
-9+7=-2
so the answer is false
Three salesmen work for the same company, selling the same product. And, although they are all paid on a weekly basis,each salesman earns his paycheck differently. Salesman A works strictly on commission. He earns $65 per sale, with a/maximum weekly commission of $1,300. Salesman B earns a weekly base salary of $300, plus a commission of $40 persale. There are no limits on the amount of commission he can earn. Salesman C does not earn any commission. His weeklysalary is $900.This task build on important concepts you've learned in this unit and allows you to apply those concepts to a variety ofsituations. Three salesmen work for the same company, selling the same product. And, although they are all paid on a weekly basis,each salesman earns his paycheck differently. Salesman A works strictly on commission. He earns $65 per sale, with amaximum weekly commission of $1,300. Salesman B earns a weekly base salary of $300, plus a commission of $40 persale. There are no limits on the amount of commission he can earn. Salesman C does not earn any commission. His weeklysalary is $900. Three salesmen work for the same company, selling the same product. And, although they are all paid on a weekly basis,each salesman earns his paycheck differently. Salesman A works strictly on commission. He earns $65 per sale, with amaximum weekly commission of $1,300. Salesman B earns a weekly base salary of $300, plus a commission of $40 persale. There are no limits on the amount of commission he can earn. Salesman C does not earn any commission. His weeklysalary is $900.
Answers
Given:
Salesman A earn = 65 per sale.
Salesman B earn = 40 per sales and 300weekly salary.
Salesman C earn = 900 weekly salary
Let "x" represent the number of sales each man
Salesman A earn is:
[tex]y=65x[/tex]Salesman B earn is:
[tex]y=40x+300[/tex][tex]\begin{gathered} 65x=40x+300 \\ 65x-40x=300 \\ 25x=300 \\ x=\frac{300}{25} \\ x=12 \end{gathered}[/tex]So total sales is 12 then.
S=0 For Zero week
[tex]\begin{gathered} \text{Salesman A} \\ y=65x \\ y=0 \end{gathered}[/tex][tex]\begin{gathered} \text{Salesman B} \\ y=40x+300 \\ y=40(0)+300 \\ y=300 \end{gathered}[/tex][tex]\begin{gathered} \text{ Salesman C} \\ y=900 \end{gathered}[/tex]For S=1
[tex]\begin{gathered} \text{Salesman A:} \\ y=65x \\ y=65(1) \\ y=65 \\ \text{Salesman B}\colon \\ y=40x+300 \\ y=40(1)+300 \\ y=340 \\ \text{Salesman C:} \\ y=900 \end{gathered}[/tex]For s=10
[tex]\begin{gathered} \text{Salesman A:} \\ y=65x \\ y=65(10) \\ y=650 \\ \text{Salesman B:} \\ y=40x+300 \\ y=40(10)+300 \\ y=700 \\ \text{Salesman c:} \\ y=900 \end{gathered}[/tex]A scale drawing for an apartment is shown below. In the drawing, 3cm represents 5m.
7,7,3,6
Answers
Actual area of the real back yard is 96 [tex]m^{2}[/tex]
[scale]=[drawing]/[real] —-> [real]=[drawing]/[scale]
scale=4 m/3 cm —> 0.75 cm/m
The backyard's measurements in the drawing are 9 x 6 cm.
Determine the backyard's actual dimensions
[Real]=[Drawing]/[Scale] for 9 cm
[real]=[9]/[0.75]——> 12 m
[Real]=[Drawing]/[Scale] for 6 cm
[real]=[6]/[0.75]——> 8 m
The backyard's actual measurements are 8 m × 12 m, and its actual size is 8 * 12 = 96 [tex]m^{2}[/tex].
Therefore the original area of the rectangle is 96 [tex]m^{2}[/tex].
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The complete question is as follows:
Below is a scale representation of a piece of land. In the illustration, 3 cm stands in for 4 m. Find the actual backyard space assuming the back is rectangular?
A police car traveling south toward Sioux Falls, Iowa, at 160km/h pursues a truck east away from Sioux Falls at 140 km/h. At time t=0, the police car is 60km north and the truck is 50km east of Sioux Falls. Calculate the rate at which the distance between the vehicles is changing at t=10 minutes, (Use decimal notation. Give your answer to three decimal places)
Answers
The rate of change of the distance between the police car and the truck after 10 minutes is approximately 193.66 m/s
What is a rate of change of a function?The rate of change of a function is the rate at which the output of the function is changing with regards to the input.
The velocity of the police car = 160 km/h south
The velocity of the truck = 140 km/h east
Distance of the police car from Sioux Falls = 60 km north
Distance of the truck from Sioux Falls = 50 km east
Required;
The rate at which the distance between the vehicles is changing at t = 10 minutes
Solution;
Let d represent the distance between the vehicles, we have;
d² = x² + y²
Where;
x = The distance of the truck from Sioux falls
y = The distance of the police car from Sioux Falls
Which gives;
[tex] \displaystyle{ \frac{d}{dt} d^2= \frac{d}{dt}(x^{2}) +\frac{d}{dt} (y^{2}) }[/tex]
Which gives;
[tex] \displaystyle{ 2 \cdot d \cdot \frac{d}{dt} d=2 \cdot x \cdot \frac{dx}{dt} + 2 \cdot y \cdot\frac{dy}{dt} }[/tex]
After 10 minutes, we have;
y = 60 - (10/60)×160 = 100/3
x = 50 + (10/60)×140 = 220/3
d = √((100/3)² + (220/3)²) = 20•(√146)/3
[tex] \displaystyle{ \frac{d}{dt} d=\frac{2 \cdot x \cdot \frac{dx}{dt} + 2 \cdot y \cdot\frac{dy}{dt} }{2 \cdot d }}[/tex]
Which gives;
[tex] \displaystyle{ \frac{d}{dt} d= \frac{2 \times \frac{220}{3} \times 140 + 2 \times \frac{100}{3} \times 160}{2 \times \frac{20 \times \sqrt{146}}{3}}}[/tex]
[tex] \displaystyle{ \frac{2 \times \frac{220}{3} \times 140 + 2 \times \frac{100}{3} \times 160}{2 \times \frac{20 \times \sqrt{146}}{3}}\approx 193.66}[/tex]
Therefore;
[tex] \displaystyle{ \frac{d}{dt} d \approx 193.66}[/tex]
The rate of change of the distance between the vehicles with time, [tex] \displaystyle{ \frac{d}{dt} d}[/tex] after 10 minutes is approximately 193.660 m/sLearn more about rate of change of a function here:
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A rectangle's base is 5 cm longer than its height. The rectangle's area is 50 cm². Find this rectangle's dimensions.
The rectangle's height is
The rectangle's base is
Answers
Answer:
The rectangle's height is 5 cm,The rectangle's base is 10 cm.Step-by-step explanation:
Let the dimensions be:
Base = b, Height = hGivenb = h + 5,A = 50 cm²The area of rectangle is:
A = bh h(h + 5) = 50h² + 5h = 50h² + 5h - 50 = 0h² - 5h + 10h - 50 = 0h(h - 5) + 10(h - 5) = 0(h - 5)(h + 10) = 0h - 5 = 0 and h + 10 = 0h = 5 and h = - 10The second value of h is negative and therefore discarded.
So we get:
h = 5 cm,b = 5 + 5 = 10 cmim having a little trouble understanding the commutative property and closure property, not a specific problem just would like a small explanation thanks!
Answers
Commutative property:
For addition and multiplication if you change the order of the addernds or factor is doesn't change the sum or multiplication.
Example:
Addition
[tex]\begin{gathered} a\pm b=b\pm a \\ \\ 5+3=3+5 \\ 8=8 \end{gathered}[/tex]Multiplication:
[tex]\begin{gathered} a\times b=b\times a \\ \\ 5\times3=3\times5 \\ 15=15 \end{gathered}[/tex]Closure property:
When you add or subtract any two intergers, the result will always be an interger.
Example:
[tex]\begin{gathered} a+b=c \\ \\ a,b,c\text{ are intergers} \\ \\ 8+13=21 \\ \end{gathered}[/tex]8, 13 and 21 are intergers
i want to ask a question
Answers
We got to use the Secant-Tangent Theorem, here. It says that AC² is equal to the distance BC times the distance from C to the circumference. Let's denote the point in OC where it intercepts the circumference by D, so:
[tex]\begin{gathered} AC^2=CD\cdot BC \\ 20^2=CD\cdot BC \end{gathered}[/tex]Beucase the radius is equal in all points of the circumference, OA=OB=OD, so BC = OA + OC, and CD = OC - OA. So,
[tex]\begin{gathered} 20^2=CD\cdot BC=(OC-OA)\cdot(OA+OC)=OC^2-OA^2=OC^2-8^2 \\ 20^2=OC^2-8^2 \\ OC^2=20^2+8^2=464 \\ OC=\sqrt[]{464}=21.54\approx22\operatorname{cm} \end{gathered}[/tex]