The inverse function of f(x) = 5x - 3 is found by switching x and y to solve for x, resulting in the inverse function f⁻¹(x) = (x + 3) / 5. This process is part of understanding function transformations that change a function's graph.
Finding the Inverse Function
Find the inverse function of f(x) = 5x - 3. To find the inverse, we need to switch the roles of x and y. So we first write y = 5x - 3. We then solve this equation for x to get x as a function of y. Performing the steps:
Add 3 to both sides: y + 3 = 5x.
Divide both sides by 5: (y + 3) / 5 = x.
This new equation x = (y + 3) / 5 represents the inverse function. To use the proper notation, we replace y with f⁻¹(x) and x with y, giving us f⁻¹(x) = (x + 3) / 5.
Understanding Function Transformations
When considering transformations such as f(-x) or f(2x), these changes to the independent variable result in specific alterations to the graph of the function, such as reflections across the y-axis or horizontal shrinkage by half.
Your principal surprises you by buying your class a turtle. He brings the turtle to your class along with a sheet from the pet store titled "Turtle Tank Rule."
*Tank walls must be at least 1 foot tall so the turtle can't climb out.
*There must be at least 400 square inches of floor space for the turtle to walk around on.
Your teacher says the volume of the tank must be smaller than 5000 cubic inches so it doesn't take up too much room in the classroom.
Give the dimensions of a tank that would work for your new turtle. Use words and numbers to explain how your tank satisfies the "Turtle Tank Rules" and your teacher requirements.
Volume of a rectangular prism= length X width X height
To satisfy the 'Turtle Tank Rules' and your teacher's requirements, the tank dimensions must meet certain criteria. You can use the formula for the volume of a rectangular prism to find the dimensions.
Explanation:To satisfy the 'Turtle Tank Rules' and your teacher's requirements, the tank dimensions must meet the following criteria:
The tank walls must be at least 1 foot tall.The tank floor must have a minimum of 400 square inches of space.The volume of the tank must be smaller than 5000 cubic inches.Let's assume the tank has length, width, and height as its dimensions. By using the formula for the volume of a rectangular prism (Volume = length * width * height), we can manipulate it to find the dimensions:
Choose a reasonable value for the height (for example, 6 inches).Calculate the area of the tank floor by dividing 400 square inches by the chosen height.Find the length and width that multiply to give this area while satisfying the volume requirement of less than 5000 cubic inches.Ethan made 14 out of 25 foul shots at basketball practice what percent of shots did he make
Which functions represent the arithmetic sequence 8, 1.5, –5, –11.5 . . . ? Check all that apply.
f(n) = –6.5n + 14.5
f(n) = –1.5n + 9.5
f(n) = 6.5n + 1.5
f(1) = 8, f(n + 1) = f(n) – 6.5
f(1) = 8, f(n + 1) = f(n) – 1.5
f(1) = 8, f(n + 1) = f(n) + 6.5
Answer:
f(n) = –6.5n + 14.5
f(1) = 8, f(n + 1) = f(n) – 6.5
Step-by-step explanation:
The nth term of an arithmetic sequence is given as
Tn = a + (n-1)d
where a is the first term and d is the common difference between two consecutive terms
From the given sequence,
a = 8, d = 1.5 - 8
d = -6.5
As such, Tn = f(n) = 8 + (n-1)-6.5
= 8 - 6.5n + 6.5
= 14.5 - 6.5n
f(1) = 8
f(n + 1) - f(n) = – 6.5
f(n + 1) = f(n) – 6.5
HELP ASAP!!!!!!
Solve and graph the inequalities below.
Describe your graphs using words as illustrated in the sample below.
SAMPLE: Part 1: The solution is x > 5 and x<=9.
Part 2: There would be an open circle on 5 and a closed circle on 9.
Part 3: The shading is between the two numbers.
Be sure that you have included all three parts when answering each question. "Greater than or equal to" would be typed >=. "Less than or equal to" would be typed <=.
1.) d - 1 > -3 and d + 2 < 5
2.) -3k + 7 > -5 and -7k + 5 < -2
3.)-5 < 2y - 3 < 23 (Hint: refer to example 2 in the lesson.)
4.)x - 4 < 5 or x + 2 > 15
5.) 5x - 4 > 6 or -4x + 5 > 1
pls help soon:)
James threw a flying disc from a height of 4 feet above the ground. The disc travels a horizontal distance of 30 feet before falling to the ground. The height of the disc, in feet, at a distance of t feet from James, is modeled on the graph below.
FILL IN BLANK
The domain represents the A. horizontal distance covered by B. maximum height of C. possible heights of the flying disc. The range of this function is [A. 1 B. 0 C. 2, A. 7 B. 6 C. 5 ]. For this function, the range represents the flying disc's A. possible heights B. maximum heights C. horizontal distance covered
hey can you please help me posted picture of question
Where does the graph of the function y=tan(x)
have asymptotes?
The graph of the function y=tan(x) has vertical asymptotes at odd multiples of pi/2 and periodic asymptotes at even multiples of pi.
The graph of the function y=tan(x) has asymptotes at x-values where the tangent function is undefined. The tangent function is undefined at x-values that are odd multiples of pi/2, such as -pi/2, pi/2, -3pi/2, etc. These x-values are called vertical asymptotes. The graph also has periodic asymptotes at x-values that are even multiples of pi, such as 0, pi, -pi, 2pi, etc. These x-values are called horizontal asymptotes.
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Which expression is equivalent to 24 - (-6), minus, left parenthesis, minus, 6, right parenthesis?
Find the critical value za/2 that corresponds to a 82% confidence level.
The critical value (or z-score) that corresponds to a 82% confidence level is approximately ±1.34, which is determined using a standard normal distribution table related to the value calculated from α/2 = 0.09.
Explanation:To find the critical value za/2 that corresponds to a 82% confidence level, we start by knowing that confidence level (CL) is the area in the middle of the standard normal distribution. We know that CL = 1 - α, hence for this particular question, α = 1 - 0.82 = 0.18, and α/2 = 0.09. The Z score associated with α/2 = 0.09 in the standard normal distribution table, which represents the critical value za/2, is approximately ±1.34.
So for an 82% confidence level, the critical value za/2 is roughly ±1.34.
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Every 6 months Reuben lopez puts 420$ into an account paying 10% compounded semiannually. Find the account balance after 15 years.
The above question is related to the future value of an amount saved at regular intervals.
We need to find the future value of the $ 420 saved every six month which are compounded semi-annually.
When the compounding is semi-annually, the same is divided by 2 and the time is multiplied by 2.
The future value will be calculated as -
Future value = $ 420 × FVIFA for 30 years @ 5 %
Here, the annual rate was 10 % but since compounding is semi-annually, thus it is divided by 2. Similarly, time is multiplied by 2 i.e 15 years X 2 = 30 periods
Referring the FVIFA table, we got -
FVIFA for 30 years @ 5 % = 66.4388
Future value = $ 420 × 66.4388 = $ 27,904.30
Which set of coordinates, when paired with (3, −3) and (10, −3), result in a square? (−3, −3) and (−10, −3) (3, 10) and (10, 3) (−3, 4) and (−10 , 4) (3, 4) and (10, 4)
The Volume of the cone shown is 6 cubic inches. What is the height of a cone with the same base diameter but a volume of only 3 cubic inches?
Answer:
3 cubic inches
Step-by-step explanation:
Using the formula, volume of the cone.
Subst. the known value of into the equation.
Thus, the volume of cone= 1(3)= 3 cubic inches.
Mario cut a circular pizza into 9 equal slices. he put a slice of pizza on each of 5 plates. what is the measure
for the angle of the slices that are left
The measure for the angle of the slices that are left is 160°.
What is multiplication?Multiplication is a mathematical arithmetic operation. It is also a process of adding the same types of expression for some number of times.
Example - 2 × 3 means 2 is added three times, or 3 is added 2 times.
Given:
Mario cut a circular pizza into 9 equal slices.
Angle of a circle = 360°
If Mario cut into 9 pieces.
360° / 9 = 40°
That means, the angle measure of each slice is 40°.
And he put a slice of pizza on each of 5 plates.
That means, he left 9 - 5 = 4 slice on the plate.
So, the angle measure;
4 x 40° = 160°
Therefore, the angle measure of the 4 slices are 160°
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Credit card A offers an APR of 27.29%, compounded monthly, while credit card B offers an APR of 27.12%, compounded daily. All else being equal, which card offers the better deal for the consumer?
The effective interest rate, also known as the effective annual interest rate, annual equivalent rate, or simply effective rate. The card which offers a better deal to the customer is Credit card A.
What is an effective rate of interest?
The effective interest rate, also known as the effective annual interest rate, annual equivalent rate, or simply effective rate, is the interest rate on a loan or financial product that is restated from the nominal interest rate and expressed as the equivalent interest rate if compound interest was payable annually in arrears. It is calculated by the formula,
EIR = [1+(i/n)]ⁿ - 1
The effective rate of interest on Credit Card A is,
Effective rate = [1+(i/n)]ⁿ - 1 = [1+(0.2729/12)]¹² - 1 = 30.97%
The effective rate of interest on Credit Card B is,
Effective rate = [1+(i/n)]ⁿ - 1 = [1+(0.2712/365)]³⁶⁵ - 1 = 31.14%
Since the effective rate of Credit card A is less than credit card B, therefore, the card which offers a better deal to the customer is Credit card A.
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What is the solution to the expression –2 – 3 – (–4)?
100 POINTS
What is the area of the figure:
Answer:
144 ft²
Step-by-step explanation:
There are a number of ways you can solve this problem. One is to figure the leg lengths, and use those to compute the area of the triangle in the usual way.
AC = (24 ft)·sin(45°) = 12√2 ft
BC = (24 ft)·cos(45°) = 12√2 ft
Area = 1/2·AC·BC = (1/2)·(12√2 ft)·(12√2 ft) = 144 ft²
___
Another is to recognize that the altitude from the hypotenuse of this isosceles right triangle is half the length of the hypotenuse, or 12 ft. Then the triangle area formula gives the area as ...
Area = (1/2)·AB·(AB/2) = (1/2)·(24 ft)·(12 ft) = 144 ft²
___
Along the same lines, the triangle is half a square whose diagonals are both 24 ft. The area of the square is half the product of those diagonals:
square area = (1/2)·(24 ft)² = 288 ft²
The triangle area is half of that, or
Area = (1/2)(288 ft²) = 144 ft²
Answer:
144 ft²
Step-by-step explanation:
AC = (24 ft)·sin(45°) = 12√2 ft
BC = (24 ft)·cos(45°) = 12√2 ft
Area = 1/2·AC·BC = (1/2)·(12√2 ft)·(12√2 ft) = 144 ft²
I need help on this question
2. A,B,C,D ?
3. A,B,C,D ?
A single fair die is tossed. find the probability of rolling a number greater than 55.
If a 10-pound sack of potatoes costs $6.55, how much would 1 pound cost? Round your answer to the nearest cent.
Given the following table, determine the missing value. Growth Factor 1.13 ? 1.28 1.97 Percent 13% 256% 28% 97%
a.
256
c.
3.56
b.
2.56
d.
35.6
Answer:
missing value in the table is 3.56
Step-by-step explanation:
Percent 13% 256% 28% 97%
Growth Factor 1.13 ? 1.28 1.97
To find growth factor first we remove the percentage and then we add 1
we are given with 246 %
to remove percentage we divide by 100
[tex]\frac{256}{100} = 2.56[/tex]
To get growth factor we add 1
1+2.56= 3.56
so growth factor is 3.56
missing value in the table is 3.56
Write 760% as a decimal and as a simplified fraction
7.6
760/100
76/10
7 6/10
760% can be expressed as a decimal by simply dividing by 100, resulting in 7.6. The same percent can be expressed as a simplified fraction by writing it over 100 and simplifying, resulting in 38/5.
Explanation:To convert 760% into a decimal and a simplified fraction, we follow the process of converting percents. First, to convert a percent to a decimal, we need to divide the percent by 100. So 760% becomes 7.6 when expressed as a decimal.
Then, to convert the percent to a simplified fraction, we write the percent as a fraction over 100. So, 760% would be written as 760/100. But it can be simplified. Both 760 and 100 can be divided by 20, so when simplified the fraction becomes 38/5.
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PLEASE HELP ME OUT ASAP! I WOULD REALLY APPRECIATE IT AND GOD BLESS
Jeff put a small box that is 12 inches long, 8inches wide, and 4 inches tall inside a box that is 20 inches long, 15 inches wide and 9 inches high. How much space is left in the larger box?
Select the best answer for the question.
2. Express the fractions 3/4, 7/16, and 5/8 with the LCD.
A. 9/16, 49/16, 36/16
B. 3/4, 2/4, 3/4
C. 12/16, 7/16, 10/16
D. 24/32, 14/32, 24/32
find the area of the regular decagon if the apothem is 2.8ft and a side is 1.8ft
Find the length of the arc shown in red. Leave your answer in terms of Pi
The calculated length of the red arc is 4.71 units
Finding the length of arc DF
From the question, we have the following parameters that can be used in our computation:
Central angle = 30 degrees
Radius = 9 meters
Using the above as a guide, we have the following:
Arc length = central angle/180 * 3.14 * Radius
Substitute the known values in the above equation, so, we have the following representation
Arc length = 30/180 * 3.14 * 9
Evaluate
Arc length = 4.71
Hence, the length of the arc is 4.71 meters
Q #4
[tex] < 6 < 1 < 2 < 4[/tex]
Someone help I don't understand this.
Devon wants to know if drinking milk before bed helps teenagers sleep. he chooses 10 friends on his high school basketball team for his study. every night for one month the friends drink a glass of milk before bed and later record how many hours they sleep that night. Describe three reasons why Devon's study design is flawed.
Answer:
The three reasons Devon's study is flawed because:
1. The sample size is very small. You cannot conclude anything based on small samples.
2. Sample is biased as Devon chose basket ball players and such players have proper sleep pattern and waking up pattern as compared to normal people. So, the study cannot be generalized.
3. Devon must have studied their sleep pattern before experimenting also, so that some quantitative data on change is available for conclusion.