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The function that is the inverse of the exponential function y = 4ˣ is lnx/ln4

The inverse of a function f is denoted by [tex]f^{-1[/tex] and it exists only when f is both one-one and onto function. Note that  [tex]f^{-1[/tex]  is NOT the reciprocal of f. The composition of the function f and the reciprocal function  [tex]f^{-1[/tex]  gives the domain value of x.

(f o  [tex]f^{-1[/tex] ) (x) = ( [tex]f^{-1[/tex]  o f) (x) = x

For a function 'f' to be considered an inverse function, each element in the range y ∈ Y has been mapped from some element x ∈ X in the domain set, and such a relation is called a one-one relation or an injunction relation.

The standard exponential function is given as y = abˣ

Given the function,

y = 4ˣ

Replace y with x

[tex]x = 4^y[/tex]

Make y the subject of the formula,

[tex]lnx = ln4^y\\\\lnx = yln4[/tex]

Divide both sides by ln4

lnx/ln4 = y

Hence the function that is the inverse of the exponential function y = 4ˣ is lnx/ln4

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Answer:

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Step-by-step explanation:

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Answer:

138ft^2

Step-by-step explanation:

9x2=18

10x12=120

18+120=138ft^2

Answer:

A: Function 1
B: Function 4
C: Function 2

Explanation A:

In order to solve this problem, you will have to find the slope for each function.

1st Function: For the first function, you will need to find 2 points, let's use (0, -4) and (1, 1). Remember slope is rise over run, so you would count the spaces from the first point to the second, it rises (y-axis) 5 points and runs (x-axis) 1 point. Which would give you 5/1 = 5. Function 1's slope is 5.

2nd Function: Same equation, rise over run, or y over x, so we need to find the pattern on each side. For the x-values, we can see starting from the first point that they increase by one, -2 to -1 to 0, and for the whole x-values, so 1 will be our run, since it is the x-value. For the rise, or y-values, we can see a pattern of -1 starting from the top going towards the bottom. Now, we can rise over run, -1/1 = -1

3rd Function: Slope is -4, as the equation is y = mx + b, and m stands for slope.

4th Function: Says slope is 2.

Now that we know all the slopes, we can see that Function 1's slope is the greatest.

Explanation B:
For this problem, we will have to find the y-intercept for all the functions, this means where the x-value is 0, so that the y-value will be right on the y-axis.

Function 1: We can see that the function intercepts the y-axis at (0, -4), it's y-intercept being -4.

Function 2: Where the x-value is 0, we can find the y-intercept, which is (0, 1). Making the y-intercept 1.

Function 3: Following the y = mx + b format, the b is the y-intercept, so it is -2.

Function 4: Gives you the y-intercept as 5.

In the end, the only function with a y-intercept greater than 4 is Function 4.

Explanation C:

Since we already know the y-intercepts from the previous problem, we only need to see which one is closest to 0. In this case, it would be Function 2 with a y-intercept at -1, only 1 unit off of 0, making it the closest.

Answer:

Step-by-step explanation:

[tex]y \geq -3x + 2\\\\y \leq x + 3[/tex]

is the equation of the given system.

Equation of the line using the coordinates (-3, 0) and (0, 3), we get:

slope = (3 - 0) / (0 - (-3)) = 1

Now we have the slope of the line. Next, we can use the point-slope form of the equation of a line, which is:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is a point on the line.

Using the point (-3, 0) and the slope we just found (which is also the slope of the line passing through the point (0, 3)), we get:

y - 0 = 1(x - (-3))

Simplifying, we get:

y = x + 3

Therefore, the equation of the line passing through (-3, 0) and (0, 3) is y=x+3.

Similarly, the equation of the line passing through (1, -1) and (0, -4) is y = -3x + 2.

Thus, from the graph the equation of the system is,

[tex]y \geq -3x + 2\\\\y \leq x + 3[/tex]

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Answer:2x-14

Step-by-step explanation:

The area of the semicircle is 47.5 square centimeters

We have to find the area of the semi circle

The diameter of the semicircle is 11 cm

Radius is half of the diameter

Radius = 5.5 cm

Let us find the area of semicircle  by formula 1/2(πr²)

Radius = 1/2×3.14×(5.5)²

= 1/2×3.14×30.25

=47.4925 square centimeters

Hence,  the area of the semicircle is 47.5 square centimeters

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Answer to the given Hypotheses is : The current shortage in the new car market has made it quicker to sell a car.

In this scenario, we have to determine if the current shortage in the new car market has affected the time it takes a car dealership to sell a new car. To answer this question, we will use the p-value approach.

First, we will determine the null and alternative hypotheses. The null hypothesis (H₀) is that the average time it takes to sell a car has not changed, while the alternative hypothesis (H₁) is that the current shortage in the new car market has made it quicker to sell a car.

H₀: μ = 20

H₁: μ < 20

Second, we will choose the appropriate test statistic and determine the level of significance. In this case, we will use a one-sample t-test, as the population standard deviation is known, and you will use a 5% level of significance (α = 0.05).

Third, we will calculate the test statistic and the p-value. The test statistic is calculated by dividing the difference between the sample mean and the hypothesized population mean by the standard error of the mean.

t = (x' - μ) / (σ / √n)

t = (15 - 20) / (12 / √30)

t = -2.89

Using a t-distribution table with 29 degrees of freedom, the p-value is 0.004.

Since the p-value is less than the level of significance, we reject the null hypothesis and conclude that the current shortage in the new car market has made it quicker to sell a car.

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With a triangle, the sum of any two side lengths must be greater than the third side length. If this is not true, then the side lengths cannot make a triangle. Let's go through each set of side lengths and determine which would and wouldn't work.

a. 3, 4, 8 - will not make a triangle

3 + 4 = 7 > 8 = false

3 + 8 = 11 > 4 = true

4 + 8 = 12 > 3 = true

b. 7, 6, 12 - will make a triangle

7 + 6 = 13 > 12 = true

7 + 12 = 19 > 6 = true

6 + 12 = 18 > 7 = true

c. 5, 11, 13 - will make a triangle

5 + 11 = 16 > 13 = true

5 + 13 = 18 > 11 = true

11 + 13 = 24 > 5 = true

d. 4, 6, 12 - will not make a triangle

4 + 6 = 10 > 12 = false

4 + 12 = 16 > 6 = true

6 + 12 = 18 > 4 = true

e. 4, 6, 10 - will not make a triangle

4 + 6 = 10 > 10 = false

4 + 10 = 14 > 6 = true

6 + 10 = 16 > 4 = true

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The outside border of the garden would have walked around approximately  58%.

Without more information about the shape and dimensions of the garden, it's impossible to give an exact answer. However, if we assume that the garden is a rectangle, we can use the formula for the perimeter of a rectangle to estimate the percentage of the garden's border that would be walked around.

Let's say that the length of the garden is L and the width of the garden is W. The perimeter of the rectangle is then:

P = 2L + 2W

If the person walks from point A to point B along the outside of the garden, they are essentially walking along two sides of the rectangle. Let's call these sides S1 and S2. Depending on the location of A and B, S1 and S2 may be two adjacent sides, two opposite sides, or one side and one diagonal.

To estimate the percentage of the garden's border that the person would walk around, we can calculate the length of S1 and S2 and divide by the total perimeter of the rectangle:

Percentage walked = (S1 + S2) / P * 100%

Again, without more information about the shape and dimensions of the garden, we can't give an exact answer. However, if we assume that the person walks along two adjacent sides of the rectangle, the percentage of the garden's border that they would walk around would be:

Percentage walked = (2L + W) / (2L + 2W) * 100%

Simplifying this expression, we get:

Percentage walked = (2L + W) / (2(L + W)) * 100%

Assuming that L and W are measured in the same units (e.g. meters), we can simplify further:

Percentage walked = (2 + W/L) / (2 + 2W/L) * 100%

For example, if the length of the garden is 10 meters and the width of the garden is 5 meters, then the percentage of the garden's border that the person would walk around if they walked along two adjacent sides would be:

Percentage walked = (2 + 5/10) / (2 + 2*5/10) * 100%

= 7/12 * 100%

= 58.3%

So the outside border of the garden would have walked around approximately  58%.

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Answer:

is every whole number a natural number

Answer:

FACTOR

(−1)×2×3×5

2 or    -150

Step-by-step explanation:

The actual length of the bus is given as follows:

45 feet.

The actual length of the bus is obtained applying the proportions in the context of the problem.

The scale drawing is of 0.5 inches to 5 feet, hence the length represented by each inch of drawing is given as follows:

5/0.5 = 10 feet.

The length of the drawing is given as follows:

4.5 inches.

Hence the actual length of the bus is given as follows:

4.5 x 10 = 45 feet.

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The given text presents a map of South Africa with various locations marked. However, important information such as the scale of the map and the number of provinces is missing. Without the scale, we cannot accurately determine distances or calculate probabilities. Similarly, the absence of information regarding the provinces prevents us from calculating the probability of selecting a specific province. Therefore, based on the given text, we can only answer the first two questions, which involve the type of scale and the general direction between two locations.

1. The type of scale given in the diagram is a ratio scale, specifically a linear scale. It represents the relationship between map distance and actual distance in a straight line.

Interpretation of the given scale:

The scale on the map is not clearly visible in the provided text. However, it is typically written as a fraction or ratio, where the numerator represents the map distance and the denominator represents the corresponding actual distance on the ground. For example, a scale of 1:1,000,000 means that 1 unit on the map represents 1,000,000 units in the real world.

2. The general direction of Badplaas from Colesburg is to the northeast.

3. To determine the actual straight-line distance between East London and Bloemfontein, we would need the scale from the map. However, the scale is not provided in the given text, making it impossible to calculate the distance accurately.

4. Stellenbosch is located in the Western Cape province on the map.

5. To calculate the probability of randomly selecting the Western Cape province in South Africa, we would need the total number of provinces in South Africa and the number of provinces that belong to the Western Cape. However, this information is not provided in the given text, so it is not possible to calculate the probability.

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The required solution of the random variable X that follows an exponential distribution is shown below.

To find the expected value, standard deviation, and probabilities associated with the exponential distribution, we'll use the given parameter λ = 1/9.848.

Expected Value (Mean):

The expected value of an exponential distribution is equal to the reciprocal of the rate parameter λ. Therefore, the expected value is given by:

E(X) = 1 / λ

E(X) = 1 / (1/9.848)

E(X) = 9.848

So, the expected value of the random variable X is 9.848.

Standard Deviation:

The standard deviation of an exponential distribution is equal to the reciprocal of the rate parameter λ. Therefore, the standard deviation is given by:

σ = 1 / λ

σ = 1 / (1/9.848)

σ = 9.848

So, the standard deviation of the random variable X is also 9.848.

P(X < 12):

To find the probability that X is less than 12, we can use the cumulative distribution function (CDF) of the exponential distribution. The CDF is given by:

[tex]F(x) = 1 - e^{(-\lambda x)}[/tex]

Substituting the given λ = 1/9.848 and x = 12, we have:

[tex]P(X < 12) = F(12) = 1 - e^{-(1/9.848) * 12)}[/tex]

Calculating this expression, we find:

P(X < 12) ≈ 0.7043

Therefore, the probability that X is less than 12 is approximately 0.7737.

P(8 < X < 12):

To find the probability that X lies between 8 and 12, we subtract the cumulative probability at 8 from the cumulative probability at 12:

P(8 < X < 12) = F(12) - F(8)
                     = [tex]1 - e^{(-(1/9.848) * 12)]} - [1 - e^{(-(1/9.848) * 8)]}[/tex]

Calculating this expression, we find:

P(8 < X < 12) ≈ 0.2025

Therefore, the probability that X lies between 8 and 12 is approximately 0.2025.

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Your ability to borrow money at lower rates improves as your credit score increases. Option D.

Your ability to borrow money at lower rates improves as your credit score increases.

A higher credit score indicates to lenders that you have a strong credit history, responsible financial behavior, and a lower risk of defaulting on loans.

Lenders view borrowers with higher credit scores as more reliable and trustworthy, making them more likely to offer lower interest rates and better loan terms.

While saving more and earning more can indirectly contribute to improving your credit score by positively impacting your financial stability and ability to manage debt, they are not direct factors that lenders consider when determining interest rates.

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Answer:

Step-by-step explanation:

Total cost function, C = 15+9x-6x²+x

By differentiation,

dc/dx= 9-12x²+1

but

dc/dx= 0

9-12x²+1 = 0

-12x = -10

x=10/12

x=5/6

All the equations which are equivalents are,

⇒ 2 + x = 5

⇒ x + 1 = 4

⇒ Negative 5 + x = negative 2

We have to given that;

All expressions are,

2 + x = 5

x + 1 = 4

9 + x = 6

x + (-4) = 7

-5 + x = -2

Now, We can simplify all the expressions as;

⇒ 2 + x = 5

⇒ x = 5 - 2

⇒ x = 3

⇒ x + 1 = 4

⇒ x = 4 - 1

⇒ x = 3

⇒ 9 + x = 6

⇒ x = 6 - 9

⇒ x = - 3

⇒ x + (-4) = 7

⇒ x = 7 + 4

⇒ x = 11

⇒ -5 + x = -2

⇒x = - 2 + 5

⇒ x = 3

Thus, All the equations which are equivalents are,

⇒ 2 + x = 5

⇒ x + 1 = 4

⇒ Negative 5 + x = negative 2

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Answer:

  10.  x = 4

  11. UT = 44

Step-by-step explanation:

You want the value of x and the length of segment UT that bisects chord PR at point U into halves marked UP = (9x -3) and UR = (7x +5). The radius of circle T is 55 units.

Point U is the midpoint of chord PR, so ...

  UP = UR

  9x -3 = 7x +5

  2x = 8 . . . . . . . . . add 3-7x

  x = 4 . . . . . . . . divide by 2

Segments UP, UT and TP form a right triangle with one leg ...

  UP = 9(4) -3 = 33

and hypotenuse 55.

We observe that the ratio of these measures being 33/55 indicates triangle PUT is a 3-4-5 right triangle, with scale factor 11. That means UT is 4·11 = 44 units.

  UT = 44 units

__

Additional comment

Recognizing triangles made from Pythagorean triples, such as {3, 4,5}, {5, 12, 13}, {7, 24, 25}, or {8, 15, 17} can save a lot of work. If you don't recognize these, you can find the missing side length from ...

  TP² = UP² +UT²

  UT = √(TP² -UP²) = √(55² -33²) = √1936 = 44

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The amount of wrapping paper which Elijah used is equal to 812 m².

In Mathematics and Geometry, the surface area of a rectangular prism can be calculated and determined by using this mathematical equation or formula:

Surface area of a rectangular prism = 2(LH + LW + WH)

Where:

By substituting the given side lengths into the formula for the surface area of a right rectangular prism, we have the following;

Surface area of a right rectangular prism = 2[3 × 10 + (10 × 12 × 2) + (13 × 12 × 2)]

Surface area of a right rectangular prism = 812 m².

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

Approx. 0.2079

Step-by-step explanation:

To find the value of sin 78°, we can use a scientific calculator or a trigonometric table. However, I can provide an approximate value using the trigonometric identity:

sin(90° - θ) = cos(θ)

In this case, we can use the identity sin(90° - 78°) = cos(78°). Since the cosine of 78° can be determined more easily, we can calculate that instead.

cos 78° ≈ 0.2079

Therefore, sin 78° is approximately equal to 0.2079.

As all sides of a Rhombus are of equal length, the length of the missing sides are 82

The property of a Rhombus required for answering this question is associated with the length of it's sides. A Rhombus has four sides with the length of all these sides being equal.

This means that with the length of one of it's sides being 82 , the length of the remaining 3 sides would be 82 as well.

The length of sides x = y = z = 82. Hence, x, y and z are 82

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Answer:

the probability that either event will occur is 0.92

Step-by-step explanation:

the probability that either event will occur is 0.92

What is the probability that either event will occur?

From the question, we have the following parameters that can be used in our computation:

Event A = 18

Event B = 15

Other Events = 3

Using the above as a guide, we have the following:

Total = A + B + C

So, we have

Total = 18 + 15 + 3

Evaluate

Total = 36

So, we have

P(A) = 18/36

P(B) = 15/36

For either events, we have

P(A or B) = 33/36 = 0.91666

Hence, the probability that either event will occur is 0.92 (rounded to nearest hundreth)

The area of the given figure is 27.5 square centimeter which has a rectangle and triangles

The given figure has a rectangle and triangles

The area of rectangle is length times width

Area of rectangle =5×4

=20 square centimeter

Area of triangle =1/2×base×height

=1/2×2.5×3

=7.5/2

=3.75 square centimeter

As there are two triangle = 2(3.75)

=7.5 square centimeter

Total area = 7.5+ 20

=27.5 square centimeter

Hence, the area of the given figure is 27.5 square centimeter

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Answer: 0

Step-by-step explanation: It is furthest away from the other numbers with a difference of at least 5. If plotted in a table it would visually be seen to be further away from the general trend.

The average rate of change for this problem is given as follows:

2.4 inches per hour.

The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function.

The change in the output is given as follows:

8.1 - 3.3 = 4.8.

The change in the input is given as follows:

2 - 0 = 2.

Hence the average rate of change is given as follows:

4.8/2 = 2.4 inches per hour.

The table is given by the image presented at the end of the answer.

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Answer

The order of the number from greatest to least is 1.62, 15/20, 16.2%

Step-by-step explanation

Given: 1.62 16.2% 15/20​

To order these data in descending order (i.e. from highest to least), follow the steps below

1. Convert all numbers to decimals

1.62

16.2% = 16.2/100

16.2% = 0.162

15/20 = 0.75

So, the number are 1.62, 0.162, 0.75

2. Compare the three decimals and arrange in descending order

1.62 > 0.75 > 0.162

This means that 1.62 is greater than 0.75 and 0.75 is greater than 0.162

By substituting 16.2% for 0.162 and 15/20 for 0.75, the order of the number from greatest to least is

1.62, 15/20, 16.2%

Answer:

The angle in the first problem measures 67°.

x=16, y=40

Step-by-step explanation:

1st problem: draw the line r in figure (red) parallel to the two given one and containing the vertex of the angle. The two green angles are congruent since alternate angles from the two parallels t and r when crossed with the upper side of the angle. Same with the two blue angles created by the parallels r and b. The measure of the angle you need is the sum of the two, or 36+31= 67°.

Second problem. The two angles in red are congruent since corresponding angles generated by the parallels m and b and the transversal r. we can then write [tex]4x-5=3x+11\\[/tex]. Solving for x we find x= 16. Now the two angles marked in blue are conjugates when looking at the parallels l and r cut by the transversal m, thus they add up to 180° We know how much the angle with double marks measures (since we found x earlier) so we can write

[tex](3y+1) + (4\times16-5) = 180\\3y+ 1 + 59 = 180\\3y=120 \implies y=40[/tex]