Two consecutive integers have a sum of 61 . Find the integers.

1. If f(x) and g(x) are inverse functions, then f(g(x)) = g(f(x)) = x. Finding the composite of the two functions will tell you if they are inverses.

2. To find the inverse of a function, swap x and y, then solve for y.

... x = 3y

... x/3 = y . . . . . matches f(x) = x/3

3. A function will pass the vertical line test. If its inverse is also a function, that, too, will pass the vertical line test. Since the inverse of a function is that function reflected across y=x, any inverse function that passes the vertical line test corresponds to an original function that passes the horizontal line test. (A vertical line reflected across y=x is a horizontal line.)

4. See 2.

... x = 3(y -2)³

... (x/3) = (y -2)³ . . . . divide by 3

... ∛(x/3) = y -2 . . . . .take the cube root

... 2+∛(x/3) = y . . . . .add 2

... f(x) = 2+∛(x/3) . . . . is the inverse

5. See 2.

... x = 2y -3

... x+3 = 2y . . . . . add 3

... (x+3)/2 = y . . . .divide by 2

... f(x) = (x+3)/2 . . . .  is the inverse

A composite function is the combination of multiple functions.

The correct answers are:

1. Test for inverse function

To test if two functions are inverse of one another, we simply find their composites.

Assume the functions are g(x) and h(x).

We simply test for [tex]g(h^{-1}(x))[/tex] and [tex]h(g^{-1}(x))[/tex]

If they are equal, then both functions are inverse functions

2. Inverse of f(x) = 3x

Rewrite as:

[tex]y = 3x[/tex]

Swap y and x

[tex]x = 3y[/tex]

Make y the subject

[tex]y = \frac x3[/tex]

Hence, the inverse function is: [tex]f'(x) = \frac x3[/tex]

3. True statements

A function has unique ordered pairs; so, it will pass the vertical line test.

Because it has unique ordered pairs, the inverse function will pass the vertical line tests, and the horizontal line tests.

Hence;

(b) All answers are correct

4. Inverse of [tex]f(x) = 3(x - 2)^3[/tex]

Rewrite as:

[tex]y = 3(x - 2)^3[/tex]

Swap x and y

[tex]x = 3(y - 2)^3[/tex]

Solve for y: Divide both sides by 3

[tex](y -2)^3 = \frac x3[/tex]

Take cube roots of both sides

[tex]y -2 = \sqrt[3]{\frac x3}[/tex]

Add 2 to both sides

[tex]y =2 + \sqrt[3]{\frac x3}[/tex]

Hence, the inverse function is: [tex]f^{-1}(x) =2 + \sqrt[3]{\frac x3}[/tex]

5. The inverse of [tex]f(x) =2x - 3[/tex]

Rewrite as:

[tex]y =2x - 3[/tex]

Swap x and y

[tex]x =2y - 3[/tex]

Solve for y: Add 3 to both sides

[tex]2y = x + 3[/tex]

Divide both sides by 2

[tex]y = \frac{x + 3}{2}[/tex]

Hence, the inverse function is: [tex]f^{-1}(x) = \frac{x + 3}{2}[/tex]

Read more about inverse functions at:

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Answer:

a. Infinitely many solutions

Step-by-step explanation:

The given equations are  2x - y = 3

4x = 6 + 2y

We can use substitution method to solve these system of equations.

2x - y = 3

y = 2x - 3

Now plug in y = 2x - 3 in the second equation, we get

4x = 6 + 2(2x - 3)

4x = 6 + 4x  - 6

4x = 4x

Here we get infinitely many solution.

Both the equations are the same.

Thank you.

See the attachment

The solid dot on the right-hand portion of the curve means the function is defined for x ≥ 2. Choices A and C have that condition.

The function is linear for x ≥ 2, so matches selection C, not A.

Answer:

About 2614 years.

Step-by-step explanation:

We are given k, and the information that N/N0 = 0.77, so the C-14 function becomes:

[tex]N=N_0\cdot e^{-kt} \\0.77N_0 = N_0\cdot e^{-kt}\\0.77 = e^{-0.0001 t}[/tex]

and we can solve for t:

[tex]0.77 = e^{-0.0001 t}\\\ln0.77 = \ln e^{-0.0001 t}\\\ln 0.77 = -0.0001 t\implies\\t = -\frac{\ln0.77}{0.0001}=2613.65\approx 2614\,\,\,\mbox{years}[/tex]

The estimated age of the bird skeleton is 2614 years.

Answer:

y = 5x+15

Step-by-step explanation:

The cost per calculator is 5

The shipping cost is 15

Let y = total cost

Let x = number of calculators

y = 5x+15

6/100 = 0.06

43/100 = 0.43

3/10 = 0.3

23/1,000 = 0.023

Hope this helps!

Final answer:

To convert fractions to decimals, divide the numerator by the denominator and combine with any whole number for mixed numbers, resulting in 0.06, 0.43, 0.3, and 4.023 respectively for the given fractions.

Explanation:

The question asks to change fractions to decimals. Here are the step-by-step conversions:

a. 6⁄100: This fraction means 6 divided by 100, which is 0.06.

b. 43⁄100: Similarly, 43 divided by 100 is 0.43.

c. 3⁄10: Dividing 3 by 10 gives you 0.3.

d. 4 23⁄1,000: For the mixed number, you have the whole number 4 and the fraction 23⁄1,000. The fraction part is 23 divided by 1,000, which is 0.023. So, you combine the whole number and the decimal to get 4.023

Converting fractions to decimals involves dividing the numerator by the denominator and adding any whole number part if it's a mixed number.

Answer:

k = -3

Step-by-step explanation:

Simplify, divide by 9, add the opposite of the constant.

0 = 9k -9·2/3 +33

0 = 9k +27

0 = k +3

-3 = k

We can use a modified form of the Pythagorean Theorem to find the length of  x, also known as side b.

Pythagorean Theorem:

a^2 + b^2 = c^2

We can fill in the values of a^2 and c^2, and then solve for b.

14^2 + b^2 = 25^2

196 + b^2 = 625

Subtract 196 from both sides.

b^2 = 429

√ both sides.

b = 20.7

Answer:

4

Step-by-step explanation:

The formula for a rectangle is base times height so if one side is 7 inches then you need to figure out what times seven equals 28, which would be four.

If one pair of sides is 7 inches and the area is length times width then the answer would be whatever you multiple by 7 to get 28, so it would be 4 inches for each of the other two sides.

What is 45% of 36 feet?

To find the percent of a number, you have to change the percent into a decimal or a fraction and multiply by the number.

DECIMAL

To change a percent into a decimal, remove the percent sign and move the decimal point 2 places to the left.

[tex]45 \% \rightarrow 0.45[/tex]

Now that you have your decimal, multiply by 36:

[tex]0.45 \times 36 = 16.2[/tex]

ANSWER = 16.2 feet

FRACTION

To change a percent into a fraction, make the percent number as the numerator(top number) and 100 as the denominator(bottom number) since fractions are out of 100.

[tex]45 \% \rightarrow \frac{45}{100}[/tex]

Now that you have your fraction, multiply by 36:

[tex]\frac{45}{100} \times 36 = \frac{1620}{100} \rightarrow 16.2[/tex]

ANSWER = 16.2 feet

If you have any questions, feel free to ask in the comments! :)

Answer:

Step-by-step explanation:

36feet=100%

18feet=50%

48.2=45%

Answer:

The side lengths are 4, 5, and 6.

Step-by-step explanation:

Each midsegment is half the length of the parallel side, so the side lengths of ΔMNK are 4, 5, and 6.

It isn't clear which point is the midpoint of what segment. If it is true that ...

then ...

Answer:

4, 5, 6

Step-by-step explanation:

Answer:

A=6

B=8

Step-by-step explanation:

The hypotenuse QR is twice the length of BC, so PQ will be twice the length of AB, 2·3 = 6; and PR will be twice the length of AC, 2·4 = 8.

In similar triangles PQR and ABC:

Side PQ (a) is 40/3 units.

Side PR (b) is 50/3 units.

To find the unknown side lengths in similar triangles PQR and ABC, we can use the properties of similar triangles. Two triangles are similar if their corresponding angles are congruent, and their corresponding sides are in proportion.

In this case, we have triangles PQR and ABC:

PQR:

RQ = 10

QP = a

RP = b

ABC:

AB = 3

AC = 4

BC = 5

Since the triangles PQR and ABC are similar, the ratios of corresponding sides must be equal. Specifically, the ratio of the sides in triangle PQR to the sides in triangle ABC should be the same. We can set up proportions to solve for a and b:

RQ / AB = QP / AC = RP / BC

10 / 3 = a / 4 = b / 5

Now, we can solve for a and b separately.

From the first part of the proportion:

10 / 3 = a / 4

Cross-multiply:

10 * 4 = 3 * a

40 = 3a

Now, divide by 3 to solve for a:

a = 40 / 3

From the second part of the proportion:

10 / 3 = b / 5

Cross-multiply:

10 * 5 = 3 * b

50 = 3b

Now, divide by 3 to solve for b:

b = 50 / 3

So, the unknown side lengths are:

a = 40/3

b = 50/3

Therefore, in similar triangles PQR and ABC:

Side PQ (a) is 40/3 units.

Side PR (b) is 50/3 units.

for such more question on triangles

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Answer:

1/8

Step-by-step explanation:

When there are two possiblities for each of 3 positions, the number of possible codes is 2^3 = 8. Your code is one of those, so its probability of occurrence is 1/8.

_____

CCC, CCP, CPC, CPP, PCC, PCP, PPC, PPP

Answer: [tex]\dfrac{1}{8}[/tex]

Step-by-step explanation:

Given: A three-character code uses the letters C and P.

If repetition is allowed then the total number of ways to make the three letter character code using two letters C and P is given by :-

[tex]2\times2\times2=8[/tex]

Since the PCP is one of the character code, therefore , the probability of the code PCP is given by :-

[tex]\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}\\\\=\dfrac{1}{8}[/tex]

Hence, the probability of the code PCP = [tex]\dfrac{1}{8}[/tex]

Answer:

None, the answer will be the same

Step-by-step explanation:

The mode is the number in the set that appears the most.

In the given data set, the number 45 is listed twice while all the other numbers are only listed once.

The mode is 45.

Answer:

45

Step-by-step explanation:

The value which appears most often or has the highest frequency in a data set is said to be the mode.

Here we are given the following data set:

{41, 43, 45, 3, 11, 23, 24, 27, 29, 45, 12, 19, 22, 49, 25}

For this data set, we can see that the number / element 45 has occurred the most often which is twice. So the mode of this data set will be 45.

Answer:

[tex]\sqrt{3}[/tex] :1

Step-by-step explanation:

Ratio of height of two cylinders are 3:1

Let C2 has the height x

then height of C1 is 3x

Let r1 is the radius of C1

and r2 is the radius of C2

As given that volume of both are equal

Also we know that formula for the volume of the cylinder is

V= π r²h

for C1

V= π (r1)²h

for C2

V=π (r2)²h

As volume of both are same so equating them

π (r1)²h1 = π (r2)²h2

as h1 =3x and h2=x

putting values

π (r1)²(3x) = π (r2)²(x)

cancelling out π and x from both side of the equation

3(r1)²= (r2)²

Taking square root of both sides give

[tex]\sqrt{3(r1)^{2} }=\sqrt{(r2)^{2} }[/tex]

r1 ( [tex]\sqrt{3}[/tex]) = r2

or

r1 : r2 =  [tex]\sqrt{3}[/tex]  :1

Answer:

75 ft

Step-by-step explanation:

Here we are given a right angled triangle with a known angle of 20°, length of the hypotenuse to be 80 and we are to find the length of the base x.

For that, we can use the formula for cosine for which we need an angle and the lengths of base and hypotenuse.

[tex]cos \alpha =\frac{base}{hypotenuse}[/tex]

So putting in the given values to get:

[tex]cos 20=\frac{x}{80} \\\\x= cos 20*80\\\\x=75.17[/tex]

Therefore, the value of x is the closest to 75 ft.

Answer:

8,800

Step-by-step explanation:

400 km

1 cm

=

d

22 cm

Answer:

208

Step-by-step explanation:

basically you can plug this into a calculator, just by pressing 160 + 30(then the percent sign), followed by the equal sign. or you can do it on paper by dividing 160 by 100, which gives you 1.6. then you multiply that by 30, with leaves you with 48. and because 48 is 30% of 160, you do 48 + 160 equals 208.

To choose unit vectors, one must understand that they have a magnitude of one and are directed along the axes in space. For any point P or S, the unit vectors are (i_p, j_p, k_p) or (i_s, j_s, k_s) respectively, each with a magnitude of one.

To determine which vectors are unit vectors, you must know that a unit vector has a magnitude (length) of one and indicates direction in space. The given information states that for a point P in space, the unit vectors are (î_p, â_p, k_p). Each of these vectors has a magnitude of one and points in the direction of increasing x, y, and z coordinates, respectively. Similarly, for point S, you have (i_s, j_s, k_s), also with a magnitude of one.

The special types of unit vectors such as î (i-hat), â (j-hat), and k (k-hat) are always of magnitude one and are parallel to the x, y, and z axes, respectively. The vector dê, represents a vector of length d pointing in the positive x-direction, implying dê is also a unit vector if d equals one.

Therefore, in the context provided, all options (a), (b), (c), and (d) are correct if their vectors have a magnitude of one and adhere to the rules to be considered unit vectors.

[tex]\dfrac{13x+10}{2x^2-5x-25}[/tex]

Multiply the first fraction by (x-5)/(x-5) and the second by (2x+5)/(2x+5). Now, you have both fractions with the common denominator (2x+5)(x-5).

[tex]\dfrac{3}{2x+5}+\dfrac{5}{x-5}=\dfrac{3(x-5)}{(2x+5)(x-5)}+\dfrac{5(2x+5)}{(2x+5)(x-5)}\\\\=\dfrac{3(x-5)+5(2x+5)}{(2x+5)(x-5)}=\dfrac{3x-15+10x+25}{2x^2-5x-25}\\\\=\dfrac{13x+10}{2x^2-5x-25}[/tex]

Answer:

3

Step-by-step explanation:

if the length and width is 3, 3 times 3 is 9

Each of these sequences goes into the classification box above it.

If the difference of terms is a constant, it is arithmetic.

... (3 -(-1)) = 4 = (7 -3), so the first sequence is arithmetic

If the ratio of terms is a constant, it is geometric.

... 12/36 = 1/3 = 4/12, so the second sequence is geometric

The third sequence has neither constant differences nor constant ratios, so is neither arithmetic nor geometric.

Answer:

Step-by-step explanation:

The distance to the lookout is the sum of distances before and after the rest stop:

... "to" distance = 2 mi + 1 3/4 mi = 3 3/4 mi

The distance from the lookout is 1/2 mile shorter, so is ...

... "from" distance = 3 3/4 mi - 2/4 mi = 3 1/4 mi

Then the total hike is ...

... total distance = "to" distance + "from" distance

... = 3 3/4 mi + 3 1/4 mi = 6 4/4 mi

... total distance = 7 mi

The water each hiker will bring is ...

... (7 mi) × (1/4 gal/mi) = 7/4 gal = 1 3/4 gal

Each hiker will hike a total of 7 miles, and they will need to bring 1.75 gallons of water for the entire trip.

First, we need to determine the total distance each hiker will travel:

The return trail is described as 1/2 mile shorter than the trail to the lookout. Therefore:

Summing up the distances:

Calculating Total Water Needed:

Each hiker brings 1/4 gallon of water per mile:

Thus, each hiker will hike a total of 7 miles and will need to bring 1.75 gallons of water for the entire trip.

(1, 1), (4, -25)

You can evaluate the function to see.

f(-1) = -3^(-1-1)+2 = -3^(-2)+2 = -1/9 +2 ≠ 2

f(1) = -3^(1-1) +2 = -1 +2 = 1

f(0) = -3^(0-1) +2 = -1/3 +2 ≠ 0

f(4) = -3^(4 -1) +2 = -27 +2 = -25

_____

Or, you can graph the points and the curve.

Answer:

(1, 1), (4, -25)

Step-by-step explanation:

12%

Her discounted price was 100% - 20% = 80% of the original price. The service fee made it be 100% + 10% = 110% of that, so her final price was ...

... 110% × 80% = 88%

of the original ticket price.

This represents a discount of 100% - 88% = 12% from the original.

Answer:

February

Step-by-step explanation:

February is colder by 7°F

-9 - 7 = -16

~

Answer:February

Step-by-step explanation:

If you see the number line below -16 is more far than -9

-------------------------------------------------------------------------------------------------

' ' '

-16 -9 0

(x, y) = (6, -6)

Divide the first equation by 2 to get equal and opposite coefficients for x.

... -2x -y = -6

Add this to the second equation to eliminate x.

... (2x +4y) +(-2x -y) = (-12) +(-6)

... 3y = -18 . . . . . . . . simplify

... y = -6 . . . . . . . . . . divide by 3

Substitute this vaue into any of the equations to find x. We choose to use the reduced first equation above.

... -2x -(-6) = -6

... -2x = -12 . . . . subtract 6

... x = 6 . . . . . . . divide by -2

The solution to the system is (x, y) = (6, -6).

Question

classify what type of angle this is

Answer:

Step-by-step explanation:

2 equal side and 2 equal angles, it is an isosceles.

Answer:

35 x 10.6, 140 x 2.65 and 371 x 1

Step-by-step explanation: