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3 years ago

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).

Mathematics

2 answers:

GrogVix [38]3 years ago

7 0

Answer:

D. f^-1(x) = log2(x -6)
4x^2 -3 . . . . x ≤ 0
B. √(x^5) -3√(x^3) -18√x

Step-by-step explanation:

1. When you replace f(x) by x and x by y, you have

... x = 2^y + 6

The first thing you do is subtract 6; then you take the base-2 logarithm:

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

... log2(x -6) = y = f^-1(x)

You know that to get the y-term by itself, you must subtract 6. Anything else you do will operate on (x-6). Only answer choice D has that sort of construction.

2. When you swap x and y and solve for y, you have ...

... x = -1/2√(y+3)

... -2x = √(y +3) . . . . . . multiply by -2

... (-2x)^2 = y +3 . . . . . square

... 4x^2 - 3 = y = f^-1(x) . . . . subtract 3

The range of f(x) is (-∞, 0], so that is the domain of f^-1(x). That is, f^-1(x) is defined for x ≤ 0.

3. The product of the two functions is ...

... (x -6)(√x)(x +3) = (√x)(x^2 -3x -18)

<em>Every term</em> will have a factor of √x, and the coefficients will be 1, -3, -18. Only selection B matches those conditions.

Leto [7]3 years ago

3 0

Answer:

Question One: log_2(x - 6) = y

Question Two: First blank: 4 Second blank 0

Question Three: B

Step-by-step explanation:

f(x) = 2^x + 6

y = 2^x + 6 Interchange the x and y

x = 2^y + 6 Subtract 6 from both sides

x - 6 = 2^y Take the log of both sides

log(x - 6) = log 2^y The power can be multiplied by the log of 2

log(x - 6) = y log(2) Divide by the log of 2

log(x - 6)/log(2) = y This to me is the preferred answer. It allows you to calculate what the actual number is or graph it on Desmos

The answer I have given is equivalent to using base 2 as your log, but calculating that is not always easy.

log_2(x - 6) = y

Answer: D

Problem Two

You want the inverse of y = - 1/2 sqrt(x + 3) x ≥ - 3

Note: you cannot have any number less than -3 because if you do, you will be taking the square root of -x which involves complex numbers. In addition x ≥ -3 is a domain. You have to keep that in mind when doing this question.

y = - 1/2 sqrt(x + 3) Interchange the x and y
x = - 1/2 sqrt(y + 3) Multiply by - 2
-2x = sqrt(y + 3) Square both sides
(-2x)^2 = sqrt(y + 3)^2
4x^2 = y + 3 Subtract 3 from both sides
y = 4x^2 - 3

The domain and range of this is a little harder to figure out. The range of f(x) becomes the domain of f-1(x)

The range of f(x) is 0 <=y < - infinity

So the domain of f-1(x) <= 0

Answer: The first blank is 4 and the second one is 0

Graph: the graph is given you below the question. What it shows is the f(x) and f^-1(x) are symmetrical about y = x. Most inverses are. If you want to check your work, it is best to include a graph when doing inverses.

Red Original
Blue inverse
Black: y = x

Problem Three

The best way to begin this problem is to apply the associative property of multiplication and start by rewriting the givens as x^(1/2)[(x - 6)(x + 3)]. Do what is in the square brackets first.

x^(1/2) [x^2 - 6x + 3x - 18]
x^(1/2) [x^2 - 3x - 18]

Now deal with the x^(1/2) which must be multiplied by all three terms in the square brackets.

x^(1/2)*x^2: x^(2 + 1/2) = x^(4/2 + 1/2) = x^(5/2)
x^(1/2)*(-3x): - 3 x^(2/2 + 1/2) = - 3x^(3/2)
x^(1/2)*(-18): - 18x^(1/2)

Result: x^(5/2) - 3x^(3/2) - 18x^(1/2)

Answer: √(x^5) - 3√x^3 - 18√x

Answer: B

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Pani-rosa [81]

This problem can be solved using probability, the equation of the probability of an event A is P(A)= favorable outcomes/possible outcomes. The interception of two probable events is P(A∩B)= P(A)P(B).

There are 12 black marbels, 10 red marbles, and 18 white marbels, all the same size. If two marbles are drawn from the jar without being replaced.

The total of the marbles is 40.

If two marbles are drawn from the jar without being replaced, what would the probability be:

1. of drawing two black marbles?

The probability of drawing one black marble is (12/40). Then, the probability of drawing another black marble after that is (11/39) due we drawing one marble before.

P(Black∩Black) = (12/40)(11/39) = 132/1560, simplifying the fraction:

P(Black∩Black) = 11/130

2. of drawing a white, then a black marble?

The probability of drawing one white marble is (18/40). Then, the probability of drawing then a black marble after that is (12/39) due we drawed one marble before.

P(White∩Black) = (18/40)(12/39) = 216/1560, simplifying the fraction:

P(White∩Black) = 9/65

3. of drawing two white marbles?

The probability of drawing one white marble is (18/40). Then, the probability of drawing another white marble after that is (17/39) due we drawed one marble before.

P(White∩White) = (18/40)(17/39) = 306/1560, simplifying the fraction:

P(White∩White) = 51/260

4. of drawing a black marble, then a red marble?

The probability of drawing one black marble is (12/40). Then, the probability of drawing then a red marble after that is (10/39) due we drawed one marble before.

P(Black∩Red) = (12/40)(10/39) = 120/1560, simplifying the fraction:

P(Black∩Red) = 1/13

4 0

3 years ago

NEED HELP ASAP!! IF YOU ANSWER RIGHT YOU GET BRAINLIEST!!!!!

lana66690 [7]

Answer:

5x^2 +10x

Step-by-step explanation:

The area is the product of the height and width. For this exercise, it is convenient to compute the rectangle areas individually, then write their sum:

blue rectangle area = (5x)(x) = 5x^2

red rectangle area = (5x)(2) = 10x

Total area = 5x^2 +10x.

_____

If you start by writing an expression for the total area, expanding it requires you deal with the unlike terms separately anyway:

5x(x +2) = 5x(x) +5x(2) = 5x^2 +10x

7 0

3 years ago

"Eight more than twice a number" can be written as what algebraic expression?

Ainat [17]

" 8 more "....means add 8
" twice a number " means multiply the number by 2

ur expression is : 2n + 8

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4 years ago

Read 2 more answers

6x-y=-23 8x+3y=-9 Substition method (Show it step by step please!)

Tanya [424]

Answer:

solution is (- 3, 5 )

Step-by-step explanation:

given the 2 equations

6x - y = - 23 → (1)

8x + 3y = - 9 → (2)

rearrange (1) expressing y in terms of x

y = 6x + 23 → (3)

Substitute y = 6x + 23 into (2)

8x + 3(6x + 23 ) = - 9

8x + 18x + 69 = - 9

26x + 69 = - 9 ( subtract 69 from both sides )

26x = - 78 ( divide both sides by 26 )

x = - 3

substitute x = - 3 into (3)

y = (6 × - 3 ) + 23 = - 18 + 23 = 5

solution is (- 3, 5 )

5 0

3 years ago

The half-life of caffeine in a healthy adult is 4.8 hours. Jeremiah drinks 18 ounces of caffeinated

statuscvo [17]

We want to see how long will take a healthy adult to reduce the caffeine in his body to a 60%. We will find that the answer is 3.55 hours.

We know that the half-life of caffeine is 4.8 hours, this means that for a given initial quantity of coffee A, after 4.8 hours that quantity reduces to A/2.

So we can define the proportion of coffee that Jeremiah has in his body as:

P(t) = 1*e^{k*t}

Such that:

P(4.8 h) = 0.5 = 1*e^{k*4.8}

Then, if we apply the natural logarithm we get:

Ln(0.5) = Ln(e^{k*4.8})

Ln(0.5) = k*4.8

Ln(0.5)/4.8 = k = -0.144

Then the equation is:

P(t) = 1*e^{-0.144*t}

Now we want to find the time such that the caffeine in his body is the 60% of what he drank that morning, then we must solve:

P(t) = 0.6 = 1*e^{-0.144*t}

Again, we use the natural logarithm:

Ln(0.6) = Ln(e^{-0.144*t})

Ln(0.6) = -0.144*t

Ln(0.6)/-0.144 = t = 3.55

So after 3.55 hours only the 60% of the coffee that he drank that morning will still be in his body.

If you want to learn more, you can read:

brainly.com/question/19599469

7 0

3 years ago