Modeling with Composite Functions
6. A store offers a 15% discount on all items and you also have a $10 coupon. Both discounts can be applied to your purchase.
a. Write a function rule, f(x) for using the coupon
b. Write a function rule, g(x) for using the 15% discount.
c. Write a function composition rule for using the $10 coupon first and then taking the 15% discount
d. Write a function composition rule for using the 15% discount first and then taking the $10 coupon.
e. Which method gives you the lowest purchase cost?
Answers
The function rules are (a) f(x) = x - 10, (b) g(x) = 0.85x, (c) g (f(x)) = 0.85x - 8.5 and (d) f (g(x)) = 0.85x - 10
What is Function?A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
Let x be the original price of the item.
(a) You have $10 coupon.
Price of the item after applying coupon = x - 10
f(x) = x - 10
(b) There is a 15% discount.
Discounted price of the item = x - (15% × x)
= x - (0.15x)
= 0.85x
g(x) = 0.85x
(c) We have f(x) = x - 10 and g(x) = 0.85x
Price of the item when applied $10 coupon first and then taking the 15% discount is,
g (f(x)) = g(x - 10) = 0.85(x - 10) = 0.85x - 8.5
(d) Price of the item when applied 15% discount first and then taking the $10 coupon is,
f (g(x)) = f(0.85x) = 0.85x - 10
(e) We have g (f(x)) = 0.85x - 8.5 and f (g(x)) = 0.85x - 10
The price will be lower for the composite function f (g(x)) = 0.85x - 10, since there is a $10 decrease in the 0.85x.
Hence the rules for the functions are f(x) = x - 10 and g(x) = 0.85x. The composite functions are g (f(x)) = 0.85x - 8.5 and f (g(x)) = 0.85x - 10.
The price will be lower when we first apply the discount and then the coupon.
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Related Questions
Evaluate the function at the given values.
K(x) =−x+2
K(1) =
K(-2) =
K(0) =
Answers
K(2)= -(2)+ 2= 0
K (0) = -(0) +2 = 2
Q1. An industry analyst wants to compare the average salaries of two firms, both to each other and to the industry. Firm A's average salary is 93% of the industry average, Firm B's average salary is $58,000, and the industry average salary is 96% of Firm B's average salary. a. Determine the industry average salary. b. Determine Firm A's average salary. c. Express Firm B's average salary as a percentage of Firm A's average salary. Round the percentage to two decimals.
Answers
a.The Industry Average Salary is $55,680. b.The Firm A's Average Salary is $51,718.40 .c. Firm B's average salary is approximately 112.27% of Firm A's average salary.
a. To determine the industry average salary, we can use the information that the industry average salary is 96% of Firm B's average salary. Firm B's average salary is $58,000. Therefore, we can calculate the industry average salary as follows:
Industry Average Salary = 96% of Firm B's Average Salary
= 0.96 * $58,000
= $55,680
b. Firm A's average salary is stated as 93% of the industry average salary. To calculate Firm A's average salary, we can multiply the industry average salary by 93%:
Firm A's Average Salary = 93% of Industry Average Salary
= 0.93 * $55,680
= $51,718.40
c. To express Firm B's average salary as a percentage of Firm A's average salary, we can divide Firm B's average salary by Firm A's average salary and multiply by 100:
Percentage = (Firm B's Average Salary / Firm A's Average Salary) * 100
= ($58,000 / $51,718.40) * 100
≈ 112.27%
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Divide. 7/8÷(−5/12) −21/10 −10/21 −35/96
Answers
Answer: - 21/10
Step-by-step explanation:
salesperson earns $345 for selling $2300 in merchendice find the commison rate
Answers
Answer:
The commission rate is 15%
Step-by-step explanation:
commission = commission rate x sales
where the commission rate is expressed as a decimal.
In this case, the salesperson earned a commission of $345 for selling $2,300 in merchandise. Therefore, we have:
345 = commission rate x 2300
To solve for the commission rate, we can divide both sides by 2300:
commission rate = 345/2300
Simplifying this expression, we get:
commission rate = 0.15
So, the commission rate is 15%
Select all the true statements
Answers
The true statements are;
1) AB + BC = AC
Length of BC = |6 - 1|
2) ∠JMK = 50°
∠KML = 35°
How to Interpret Number Line intervals?
1) From the given number line interval, we can deduce the interpretation as follows;
AB + BC = AC
Length of BC = 5 units or |6 - 1|
Length of B = 3 Units
2) We can see that ∠JML is an angle triangle. Thus, ∠JML = 85°. Thus;
6x + 2 + 4x + 3 = 85
10x + 5 = 85
10x = 85 - 5
10x = 80
x = 80/10
x = 8°
Thus;
∠JMK = 6(8) + 2
∠JMK = 50°
∠KML = 85° - 50°
∠KML = 35°
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Calcular los 3/5 de los 2/3 de las 3/4 de 560
Answers
For the fractions, the calculation of 3/5 of 2/3 of 3/4 of 560 is equal to 168.
How to solve fractions?To calculate 3/5 of 2/3 of 3/4 of 560, break it down step by step:
Step 1: Calculate 3/4 of 560:
3/4 × 560 = (3 × 560) / 4 = 1680 / 4 = 420
Step 2: Calculate 2/3 of the result from Step 1:
2/3 × 420 = (2 × 420) / 3 = 840 / 3 = 280
Step 3: Calculate 3/5 of the result from Step 2:
3/5 × 280 = (3 × 280) / 5 = 840 / 5 = 168
Therefore, 3/5 of 2/3 of 3/4 of 560 is equal to 168.
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(h) Three meals per day are provided by the hotel. The hotel will charge a total of $9,240 for all meals for the entire group. What is the cost of meals for each person?
Answers
Answer:
$3,080 for each person
Step-by-step explanation:
9,240 / 3 = 3,080
Help anyone need it asap
8th grade math
Answers
Answer:
x = 1
Step-by-step explanation:
2x + 8 = 4x + 6
subtract 6 from both sides of the equation
2x + 8 - 6 = 4x + 6 - 6
2x + 2 = 4x
subtract 2x from both sides of the equation
2x - 2x + 2 = 4x - 2x
2 = 4x - 2x
2 = 2x
divide both sides of the equation by 2
2/2 = 2x/2
1 = x
Does anyone know how to do this?
Answers
Step-by-step explanation:
For a. We must find the lim as x approaches 2 from the right.
We can use direct substitution for the second equation because
We can direct subsitute since we not dealing with no asymptotesWe use the second equation because that is when x is greater than or equal to two so we use that because we can determine the limit.So just subsitue 2 in for x.
\( - 6(2) {}^{2} + 2m\)
\( - 6(4) + 2m\)
\(2m - 24\)
So the limit as x approaches 2 from the right, is 2m-24.
b. We now must find the limit as x approaches 2 from the left,
This time we will use the top equation because since the x values for the top equation is only defined to be less than 2, we can know the behavior as it approaches 2 from the left.
Here we direct subsitue again
\(6(2) + m\)
\(12 + m\)
So as x approaches 2 from the left, the limit is 12+m
Here we let
\(12 + m = 2m - 24\)
\(36 = m\)
The value is 36 so
m=36
Antonio buys a television worth $700 at a rent-to-own business. Because he doesn't have $700 on the day of purchase, he will pay $100 per
month for 12 months and will be able to take the television home immediately. How much will Antonio end up spending on the television after
his 12 payments?
OA. $100
OB. $500
OC. $700
OD. $1200
Answers
Answer:
$1200
Step-by-step explanation:
Since he's paying $100 on a 12 month plan, you do
100 * 12 = 1200
In total he's paying $1,200.
Answer: 1200
Step-by-step explanation: Hope this helps :)
....................help
Answers
Answer:
A
Step-by-step explanation:
With function transformation, added/subtracted numbers outside of the x always mean the function is being translated up/down. In this case, since the 2 is being replaced with a 4, the new function is 2 units higher than the previous function.
HELP PLEASE FAST!!!!!
Answers
Answer: first choice is correct
Step-by-step explanation:
\(\sqrt{10} = 3.16\\\\ \sqrt{13} = 3.6\)
22/9 = 2.44
Therefore Point A is 22/9, Point B is \(\sqrt{10}\), Point C is \(\sqrt{13}\)
the 3rd and 6th term in fibonacci sequence are 7 and 31 respectively find the 1st and 2nd terms of the sequence
Answers
Answer:
Step-by-step explanation:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144,233,377,610,987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, ... Can you figure out the next few numbers?
If a wheelchair access ramp has to have an angle of elevation no more than 4.8 degrees and it has to rise 18 inches above the ground, how long must the ramp be?
Answers
The wheelchair access ramp must be 216.09 inches long.
To find the length of the wheelchair access ramp, we can use trigonometry.
The tangent function relates the angle of elevation to the ratio of the opposite side (height) to the adjacent side (length of the ramp).
Let's denote the length of the ramp as "x".
The height of the ramp is given as 18 inches.
Using the tangent function:
tan(angle of elevation) = height/length of the ramp
tan(4.8 degrees) = 18/x
To solve for x, we can rearrange the equation:
x = 18 / tan(4.8 degrees)
Using a calculator to evaluate the tangent of 4.8 degrees:
x = 18 / 0.08331
x= 216.09
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Match each missing side length and angle with the correct value. Angle measurements are rounded to the nearest hundredth. 15 M LIE, 10 8 8 N 20 22 36.87 32.23° 12 28.07 53.13 17 6 LY MZNMY NM MZNLY Ehts reserved
Answers
SOLUTION
Now, let us split the triangle in the picture into two as shown below
Now from the diagram above, we can see the two right-angle triangles.
From the smaller triangle NLY, to find the side LY, we will use the Pythagorean theorem. From the Pythagorean theorem
\(\text{hypotenuse}^2=opposite^2+adjacent^2\)From here we have that
\(\begin{gathered} |NL|^2=|NY|^2+|LY|^2 \\ 10^2=8^2+|LY|^2 \\ 100=64+|LY|^2 \\ |LY|^2=100-64 \\ |LY|^2=36 \\ \text{square root both sides } \\ \sqrt[]{|LY|^2}=\sqrt[]{36} \\ \text{square cancels square root, then } \\ |LY|=\sqrt[]{36} \\ |LY|=6 \end{gathered}\)hence LY is 6
Also form the second triangle NMY, using Pythagoras theorem to find NM, we have that
\(\begin{gathered} |NM|^2=|NY|^2+|MY|^2 \\ |NM|^2=8^2+15^2 \\ |NM|^2=64+225 \\ |NM|^2=289 \\ \sqrt[]{|NM|^2}=\sqrt[]{289} \\ |NM|=\sqrt[]{289} \\ |NM|=17 \end{gathered}\)Hence, NM = 17
We will use the trig ratio SOHCAHTOA to find angle NMY. NMY is the marked angle in the second triangle. So so using the sides 8 (opposite) and 15(adjacent) we have
\(\begin{gathered} \tan |NMY|=\frac{opposite}{\text{adjacent}} \\ \tan |NMY|=\frac{8}{\text{1}5} \\ \text{when tan moves to the other side it becomes tan}^{-1} \\ |NMY|=\text{tan}^{-1}\frac{8}{\text{1}5} \\ |NMY|=28.07^o \end{gathered}\)Hence angle NMY = 28.07 degrees
In the first triangle to find angle NLY, using the hypotenuse 10 and the opposite 8, we have
\(\begin{gathered} \sin |\text{NLY}|=\frac{opposite\text{ }}{\text{hypotenuse}} \\ \sin |\text{NLY}|=\frac{8}{\text{1}0} \\ |\text{NLY}|=\sin ^{-1}\frac{8}{\text{1}0} \\ |\text{NLY}|=53.13^o \end{gathered}\)Hence angle NLY = 53.13 degrees
NO LINKS!! URGENT HELP PLEASE!!!
Please help me with Growth rate and Initial Value only
Answers
Answer:
growth rate: 4
y-value: 19
equation: y=4x+19
Step-by-step explanation:
Growth Rate:
The growth rate of a linear function is constant. This means that the function will increase or decrease by the same amount for every unit increase in x.
This can be found by dividing the change in y-values by the change in x-values.
For the question:
The change in y-values is 11-7=4,
and the change in x-values is +1.
Therefore, the growth rate is 4.
\(\hrulefill\)
Initial Value: The initial value of a linear function is the value of the function when x is 0.
In this case, the initial value is 19.
This can be found by looking at the y-value of the point where x is 0.
In this case, the y-value is 19.
\(\hrulefill\)Equation: The equation of a linear function is y = mx + b, where m is the slope and b is the y-intercept.
Using the table you provided, we can find the slope by using two points on the line.
Let’s use (-3, 7) and (1, 23).
The slope is (y2-y1)/(x2-x1)=(23-7)(1-(-3)=16/4=4
Now,
Taking 1 point (-3,7) and slope 4.
we can find the equation by using formula:
y-y1=m(x-x1)
y-7=4(x+3)
y=4x+12+7
y=4x+19
Therefore, the equation of the given table is y=4x+19\(\hrulefill\)
Answer:
Growth rate: 4
Initial value: 19
Equation: y = 4x + 19
Step-by-step explanation:
The slope of a linear function represents its growth rate.
Therefore, the growth rate of a linear function can be found using the slope formula.
Substitute two (x, y) points from the table into the slope formula, and solve for m. Substituting points (0, 19) and (1, 23):
\(\textsf{slope}\:(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{23-19}{1-0}=\dfrac{4}{1}=4\)
Therefore, the growth rate of the linear function is 4.
The initial value of a linear function refers to the y-intercept, which is the value of the y when x = 0.
From inspection of the given table, y = 19 when x = 0.
Therefore, the initial value of the linear function is 19.
To write a linear equation given the growth rate (slope) and initial value (y-intercept), we can use the slope-intercept formula, which is y = mx + b. The slope is represented by the variable m, and the y-intercept is represented by the variable b.
As the growth rate of the given linear function is 4, and the initial value is 19, substitute m = 4 and b = 19 into the slope-intercept formula to create the equation of the linear function represented by the given table:
\(\boxed{y=4x+19}\)
The amount of money that Ryan earns is directly proportional to the number hours that he works. How much does Ryan earn per hour?
Answers
The earnings of Rayan per hour is $5 per hour
The first point = (1, 5)
The second point = (2, 10)
The slope of the line is the change in y coordinates with respect to the change in x coordinates of the function
The slope of the line m = \(\frac{y_2-y_1}{x_2-x_1}\)
Where m is the slope of the line
\((x_1,y_1)\) is the coordinates of the first point
\((x_2,y_2)\) is the coordinates of the second point
Substitute the values in the equation
The slope of the line m = (10-5) / (2-1)
Subtract the terms
= 5 / 1
= $5 per hour
Hence, the earnings of Rayan per hour is $5 per hour
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what the dog doing ?
Answers
The dog is playing with his chew toy!
Answer:
Explanation
Step-by-step explanation:
In general, dogs can run 15-20 mph
humans can run 6-8 mph
You are significantly slower than the dog,
the fastest human is around 27 miles per hour, unfortunately, even if you were this fast
the fastest dog can run 40 miles per hour as an estimate
even house cats can run at 30 mph maximum if they really wanted,
You have no chance of outrunning this dog, and barely a chance of outsmarting it if that option was available, in conclusion, the dog is chasing you at a speed of around 30 mph about 6-7x more than your own speed with no chance of outrunning it(the dog).
PLEASE HELP DONT IGNORE!!
Answers
Answer:
\(m=-\frac{1}{2}\)
Step-by-step explanation:
\(\Delta y=-1,\,\Delta x=2\\\rightarrow \frac{\Delta y}{\Delta x}=\frac{-1}{2}=-\frac{1}{2}\)
\(\Delta\) represents change in both variables, so we calculate the slope this way.
Answer:
m=-1/2
Step-by-step explanation:
Use slope formula: \(m=\frac{y_{2} -y_{1}}{x_{2}-x_{1}}\)
There are four rational numbers which are listed smallest to largest, A, B, C, and D. If the numbers are 5/3 7/15 6/10 and 25/5 , identify which is A, which is B, which is C, and which is D. brainly
Answers
The rational numbers are given as follows:
A = 7/15.B = 6/10.C = 5/3.D = 25/5.How to order the rational numbers?To order the rational numbers, we must convert the fractions to decimal, dividing the numerator of the fraction by the denominator.
The conversion of each number is given as follows:
5/3 = 1.67.7/15 = 0.47.6/10 = 0.6.25/5 = 5.Hence they are ordered as follows:
7/15, 6/10, 5/3, 25/5.
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How many permutations of the letters PRODUCT have consonants in the second and third positions?
Answers
Answer:
there are 144 permutations of the letters PRODUCT with consonants in the second and third positions.
Step-by-step explanation:
The word PRODUCT has 6 letters, so there are 6! = 720 possible permutations of its letters.
To count the permutations where the second and third positions have consonants, we can first choose the consonants for these positions. There are 3 consonants in the word PRODUCT (P, R, and C), so we can choose 2 of them in C(3,2) = 3 ways. Once we have chosen the consonants, we can arrange them in the second and third positions in 2! = 2 ways (since there are two positions to fill).
The remaining 4 letters (O, U, and T) are vowels and can be arranged in the remaining 4 positions in 4! = 24 ways.
Therefore, the total number of permutations of the letters PRODUCT with consonants in the second and third positions is:
3 × 2! × 4! = 3 × 2 × 24 = 144
So there are 144 permutations of the letters PRODUCT with consonants in the second and third positions.
What are the two categories of fees?
Answers
Step-by-step explanation:
Types of fees and charges
Monthly account keeping or service fee. The fee you pay for an organisation to manage your bank account. ...
Internet banking fee. ...
EFTPOS transaction fee. ...
ATM transaction fee. ...
Non-bank or foreign ATM fee. ...
Telephone banking transaction fee. ...
Branch withdrawal fee. ...
Cheque withdrawal fee.
Examine the steps used to solve the equation.
Negative 3 y + two-thirds = 2 y minus 4. 1. Two-thirds = 5 y minus 4. 2. StartFraction 14 Over 3 EndFraction = 5 y. 3. (one-fifth) StartFraction 14 Over 3 EndFraction = (one-fifth) 5 y.
Evaluate the steps used to solve the equation, and then describe each step.
Step 1:
Step 2:
Step 3:
What is the solution to the equation?
y =
Answers
Answer:
can confirm guy or girl above me
Step-by-step explanation:
The solution to the equation is y=14/15.
The given equation is -3y+2/3=2y-4.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Now, transpose -3y to RHS and simplify.
That is, 2/3=5y-4
Transpose -4 to LHS and simplify.
That is, 14/3=5y
Multiply by 1/5 on both sides of the equation.
So, 14/3×1/5=5y×1/5
⇒y=14/15
Therefore, the solution to the equation is y=14/15.
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A bottle contains 2 liters of water. The bottle leaks 80 milliliters of water every 3 minutes. Will the bottle be empty in an hour
Answers
Answer:
No
Step-by-step explanation:
2 liters = 2,000
80/3 = 26.6
the bottle leaks 26.6 milliliters in a min
26.6*60 = 1596
1596 / 1000 = 1.596 (to convert to liters)
the bottle will leak 1.596 an hour
hope this helps!
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If each letter of the alphabet is worth two more than its preceding letter, where A is
the first number and equals 1, B is the second number and equals 3,... and Z is the
twenty-sixth which equals What is the value of Z?
Answers
Each letter of the alphabet is worth two times as much as the one before it, implying that the value of each letter rises in mathematical progression. The formula for finding the nth term of an arithmetic progression would be used. I am written as
a + (n - 1)d = Tn
Where
The number of terms in the arithmetic sequence is represented by n.
The common difference of the terms in the arithmetic sequence is represented by d.
The first term of the arithmetic sequence is represented by a.
Tn stands for the nth word.
Based on the facts provided,
n = 26 characters1 Equals a
3 minus 1 equals 2 (difference between 2 letters)
Therefore,
1 + (26 - 1)2 = T26
51 = T26
The formula for calculating the sum of an arithmetic sequence's n terms
is as follows:
[2a + (n - 1)d] Sn = n/2
As a result, S26 is the sum of the first 26 terms.
S26 = 20/2[2 1 + (26 - 1)2] S26 = 20/2
[2 + 50] S26 =
676 = S26 = 13 52
can I get a step by step explanation Thnx
Answers
Answer:
( 2A - kn) /k = m
Step-by-step explanation:
A = k/2(m+n)
Multiply each side by 2/k
2/k *A =2/k * k/2(m+n)
2A /k = m+n
Subtract n from each side
2A /k - n = m+n -n
2A /k - n = m
Getting a common denominator
2A/k - kn/k = m
( 2A - kn) /k = m
Answer:
Step-by-step explanation:
\(A=\frac{k(m+n)}{2}\\2A=k(m+n)\\\frac{2A}{k} =m+n\\\frac{2A}{k}-n=m\\2A-kn=km\\\frac{(2A-kn)}{k}=m\)
The length of a rectangle is 4ft less than 3 times width. Perimeter is 54 ft. What equation can be used to find the width of the rectangle
(3w-4)+2w=54
2(3w-4)+2w=54
w(3w-4)=54
2(4w-3)+2w=54
Answers
Answer:
b
Step-by-step explanation:
Here,
The length of a rectangle is 4ft less than 3 times the width
Then,
length=3w-4ft
Now,
2(l+w)=P
2(3w-4+w)=54
6w-8+2w=54
2(3w-4)+2w=54
Equation b can be used to find the width of the rectangle
what is y-intercept of this problem
Answers
Answer:
the y-intercepts (s) represents the value of y(x) when x = 0. on an x,y graph, a Y-intercept is the part or the curve that crosses the y-axis.
cosider the line equation,y(x)=mx+b,a,b are constants. then, b is the Y-intercept because y(0)=m x 0 + b or y(0)=b
Step-by-step explanation:
hope it helps
Solve the equation √3 Cosecx-2=0
Answers
Answer:
60
Step-by-step explanation:
\( \sqrt{3} cosecx - 2 = 0\)
\( \sqrt{3} cosecx = 2\)
\(cosec = \frac{2}{ \sqrt{3} } \)
\(sinx = \frac{ \sqrt{3} }{2} \)
\(x = sin {?}^{ - 1} ( \frac{ \sqrt{3} }{2} )\)
\(x = 60\)
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Please help me with these Venn Diagram circles
Answers
Figures with more than 3 sides but with no pairs of parallel sides:
Kite,Pentagon,HeptagonIntersectionFigures with more than 3 sides and with one or more pairs of parallel sides, almost all even sided polygons:
Square,Parallelogram,Rhombus,Trapezoid,Rectangle,Hexagon,OctagonCircle 2Figures with one or more pairs of parallel sides:
Empty, all included into intersection regionProblem 2Circle 1Figures with less than 4 sides:
Scalene triangleIntersectionFigures with less than 4 sides and with at least two sides of equal length:
Equilateral triangle,Isosceles triangleCircle 2Figures with at least two sides of equal length:
Square,Rectangle,Rhombus,Kite,Isosceles trapezoid,Pentagon,Hexagon,Heptagon,OctagonProblem 3Circle 1Figures with at least one curved side:
CircleIntersectionFigures with at least one curved side and with at least one obtuse angle:
Empty, not considering combination of shapesCircle 2Figures with at least one obtuse angle (polygons with more than 4 sides will have obtuse angles)
Trapezoid,Rhombus,Kite,Parallelogram,Pentagon,Hexagon,Heptagon,Octagon