if f(x)= 1 over 9x -2, what is f^-1(x)?

Work Shown:

y = 5x+8

0 = 5*8+8

0 = 40+8

0 = 48

The last equation is false, so the original equation is false when x = 8 and y = 0. This means the point (8,0) is NOT found on the line.

Visual confirmation is shown below.

Answer: The two solutions are (2, 4) and (-1, 1).

Step-by-step explanation:

The given is a system of equations, y = x + 2, and, y = x^2

A system of equations comprises two or more equations and seeks common solutions to the equations.

To solve you can use substitution. And by doing so you can replace y of one equation with what the other equation equals:

y = x + 2

y = x^2

->   x + 2 = x^2

Now lets get all variables and constant to one side of the equation, set it equal to zero.

x + 2 = x^2

-x -2           -x -2

x^2 -x -2 = 0

Lets factor to find the solution

Factoring is used to simplify an algebraic expression by finding the greatest common factors that are shared by the terms in the expression.

We can factor, x^2 -x -2 = 0, into:

(x - 2)(x + 1) = 0

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SIDE NOTE:

If confuse on factoring please see attached image.

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with (x - 2)(x + 1) = 0 we can find the solution by setting each component in the parentheses to 0:

x - 2 = 0

x + 1 = 0

Now you must solve for x.

x - 2 = 0

+2      +2

x = 2

x + 1 = 0

  -1      -1

x = -1

x = 2 and x = -1

However, we are not done yet, we need to find what y is, and by doing so we can plug in the x values we got to find the corresponding y value to create points (coordinates).

-

When x = 2

y = x + 2

y = (2) + 2

y = 4

(2, 4)

-

When x = -1

y = x + 2

y = -1 + 2

y = 1

(-1, 1)

-

The two solutions are (2, 4) and (-1, 1).

To factor the expression 4x^2 - 25, we can use the difference of squares formula, which states that a^2 - b^2 can be factored as (a + b)(a - b).

In this case, we have 4x^2 - 25, which can be written as (2x)^2 - 5^2. Comparing it with the difference of squares formula, we can identify that a = 2x and b = 5. Therefore, the correct option is:

b. (2x)^2 - (5)^2

Using the difference of squares formula, we can factor it as follows:

(2x + 5)(2x - 5)

Hence, the correct factorization of 4x^2 - 25 is (2x + 5)(2x - 5), which is equivalent to option b.

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The number of subsets in the given set is as follows:

16.

Considering a set with n elements, the number of subsets in the set is the nth power of 2, that is:

[tex]2^n[/tex]

The set in this problem is composed by integers between 2 and 5, hence it has these following elements:

{2, 3, 4, 5}.

The set has four elements, meaning that n = 4, hence the number of subsets is given as follows:

[tex]2^4 = 16[/tex]

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Answer:

Step-by-step explanation:

g'(x) = 0.5*[(3x)^(-0.5)]*3 (By the power and chain rule)

And then just plug in 3, 0, and 2 into the given equation for g'(x)

6a. The equation for the graph is y = 2√(x + 5).

6b. The equation for the graph is y = -|x + 1| + 5

In Mathematics and Geometry, the standard form of a square root function can be modeled as follows;

y = a√(x - h) + k

Part 6a.

Next, we would determine value of a as follows;

4 = a√(-1 + 5) + 0

4 = a√4

4 = 2a

a = 2

Therefore, the required square root function is given by;

y = 2√(x + 5)

Part 6b.

Since the line representing the absolute value function has a y-intercept at (0, 4) and vertex at (-1, 5), the absolute value equation for the graph is given by:

y = a|x - h| + k

y = -|x + 1| + 5

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Answer: 1000

Step-by-step explanation:

If one person can pick up a plate every 10 seconds, then the rate of picking up plates can be represented as 1 plate/10 seconds.

When two people are picking up plates, the rate doubles, so it becomes 2 plates/10 seconds or 1 plate/5 seconds.

To find how long it will take for 200 people to pick up plates, we need to use the formula:

time = amount ÷ rate

Plugging in the values, we get:

time = 200 plates ÷ (1 plate/5 seconds)

time = 200 plates × 5 seconds/plate

time = 1000 seconds

Therefore, it will take 1000 seconds for 200 people to pick up plates at a faster rate.

I hope that this answer has helped you!

Answer:1 11/12

Step-by-step explanation:

Answer:

  (c)  The x-intercept of f⁻¹(x) is the same as the y-intercept of f(x).

Step-by-step explanation:

You want to know the relationship between the graphs of function f(x) and its inverse f⁻¹(x).

The inverse of a function maps every y-value of the original function to its corresponding x-value. That is if you have ...

  f(a) = b

then the graph of f(x) contains the ordered pair (a, b).

The inverse function will have the ordered pair (b, a). That is,

  f⁻¹(b) = a

If an ordered pair (x-intercept) of the inverse function is ...

  (P, 0)

Then there will be an ordered pair (0, P) on the graph of the original function. That point is the y-intercept, and its y-coordinate is the same as the x-coordinate of the x-intercept of the inverse function.

The x-intercept of f⁻¹(x) is the same as the y-intercept of f(x).

__

Additional comment

The graphs of the two functions are mirror images of each other across the line y=x.

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Answer:

The least number that should be subtracted from 1456 to make a perfect square with formula is 9^2, which is equal to 81.

Step-by-step explanation:

To find the least number that should be subtracted from 1456 to make a perfect square with formula, we need to use the formula for the square of a number.

(n + a)squared = nsquared + 2nas + asquared

In our case, since we want to find the least number to be subtracted, we need to make the result of the formula 2nas + asquared negative (so the difference is a perfect square)

2nas + asquared < 0

Let's try for some values of n:

1456 - n = aperfect square

1456 - 5^2 = (1456 - 25)^2 = 1431^2

1456 - 7^2 = (1456 - 49)^2 = 1407^2

1456 - 9^2 = (1456 - 81)^2 = 1375^2

Since 1375^2 is the smallest of the squares calculated so far, we can conclude that the smallest number that should be subtracted from 1456 to make a perfect square formula with a perfect square result is 1456 - 9^2 == 1456 - 81 == 9^2 == 81

So in this case, the least number that should be subtracted from 1456 to make a perfect square with formula is 9^2, which is equal to 81.

Isabella's possible numbers can be represented by the compound inequality 15 ≤ 5n - 5 ≤ 50, which in interval notation is [4, 11].

Based on the given information, we can set up a compound inequality to represent the range of numbers that Isabella might be thinking of.

Let's denote the number Isabella is thinking of as 'n'.

The clue states that "5 less than 5 times his number is at least 15 and at most 50."

We can express this as:

15 ≤ 5n - 5 ≤ 50

To solve this compound inequality, we add 5 to all three parts of the inequality:

15 + 5 ≤ 5n - 5 + 5 ≤ 50 + 5

20 ≤ 5n ≤ 55

Finally, dividing all parts of the inequality by 5:

20/5 ≤ n ≤ 55/5

4 ≤ n ≤ 11

Therefore, Isabella's number, 'n', lies in the range [4, 11] in interval notation. In summary, the compound inequality 15 ≤ 5n - 5 ≤ 50 represents the range of numbers that Isabella might be thinking of. The interval notation [4, 11] indicates that her number could be any value between 4 and 11, inclusive.

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To prepare a July 31 Balance Sheet for Academic Learning Services, you would need access to the financial statements and records provided in the class, as well as the information from Practical Exercise

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The null and alternative hypotheses can be stated as follows:

c. H0: µ = 12, Ha: µ ≠ 12

The null hypothesis (H0) assumes that the population mean content of the bottles is 12 ounces, indicating perfect adjustment of the filling machine. The alternative hypothesis (Ha) states that the population mean content is not equal to 12 ounces, suggesting that the machine is not in perfect adjustment.

The rejection region for α = 0.01 can be specified as:

c. |t| > 2.68

This means that we would reject the null hypothesis if the absolute value of the calculated t-statistic is greater than 2.68.

To calculate the p-value, we need the t-statistic corresponding to the sample mean and standard deviation. With a sample mean of 11.88 ounces, a standard deviation of 0.35 ounces, and a sample size of 49, we can calculate the t-statistic. The p-value represents the probability of observing a sample mean as extreme as the one obtained, assuming the null hypothesis is true.

The p-value cannot be determined without the t-statistic value or the corresponding degrees of freedom.

Without the p-value, we cannot draw a definitive conclusion. To make a conclusion, we would compare the calculated t-statistic to the critical t-value based on the chosen significance level (α = 0.01). If the calculated t-statistic falls within the rejection region (|t| > 2.68), we would reject the null hypothesis. If the calculated t-statistic falls outside the rejection region, we would fail to reject the null hypothesis.

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The top left graph represents the weight of the radioactive isotope over time.

An exponential function has the definition presented according to the equation as follows:

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

In which the parameters are given as follows:

The parameter values for the function in this problem are given as follows:

a = 96, b = 0.5.

Hence the function is given as follows:

[tex]y = 96(0.5)^x[/tex]

Two points on the graph of the function are given as follows:

(1,48) and (2, 24).

Hence the top left graph represents the weight of the radioactive isotope over time.

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Answer:

Graph W

Step-by-step explanation:

The given information describes a radioactive decay process, where the weight of the isotope decreases by half at regular intervals. This type of decay is characteristic of exponential decay.

Based on the description, the graph that represents the weight of the radioactive isotope over time would be a decreasing exponential curve, where the y-axis represents the weight of the isotope (in grams), and the x-axis represents time (in hours).

The initial weight of the isotope is 96 grams, and after each subsequent hour, the weight becomes half of what it was in the previous hour. Therefore, the correct graph would start at 96 grams (the initial weight when x = 0) and then decrease by half every hour. It would be a curve that gets closer and closer to zero but never quite reaches it.

Initial weight: 96 grams

After 1 hour: 96 / 2 = 48 grams

After 2 hours: 48 / 2 = 24 grams

After 3 hours: 24 / 2 = 12 grams

After 4 hours: 12 / 2 = 6 grams

After 5 hours: 6 / 2 = 3 grams

So, the points on the graph would be:

Therefore, the graph that represents the weight of the radioactive isotope over time is Graph W.

The solution above is graphed correctly in the last option choice is x > -6

We have been given the equation 2x + 3> -9

In order to graph the solution, we must find the value of x

2x + 3 > -9

Subtract three from both sides

-9 - 3 = -12

2x > -12

Divide both sides by 2

x > -6

Examine the inequality symbol to figure out how to graph the solution. If the sign is "greater than," graph the line to the left. If it was "less than," you would draw a straight line.

We have the "greater than" symbol in our issue, which implies we will graph our line to the right, and since we start our line at -6, we know the last choice is the correct solution.

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The following question may be like this:

Which graph shows the solution set for 2x+3>-9.

The valid numbers that satisfy all the given inequalities are -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

Let's solve each inequality step by step:

1. Two less than a number (x) is no more than 5:

The inequality is given as x - 2 ≤ 5. Adding 2 to both sides, we get x ≤ 7. This means that any number less than or equal to 7 satisfies the inequality.

2. Seven subtracted from 4 times a number (x) is more than 13:

The inequality is given as 4x - 7 > 13. Adding 7 to both sides, we get 4x > 20. Dividing both sides by 4, we obtain x > 5. So any number greater than 5 satisfies the inequality.

3. The sum of two times a number (x) and -2 is at least 8:

The inequality is given as 2x - 2 ≥ 8. Adding 2 to both sides, we get 2x ≥ 10. Dividing both sides by 2, we have x ≥ 5. So any number greater than or equal to 5 satisfies the inequality.

4. Four added to 3 times a number (x) is less than 19:

The inequality is given as 3x + 4 < 19. Subtracting 4 from both sides, we get 3x < 15. Dividing both sides by 3, we obtain x < 5. So any number less than 5 satisfies the inequality.

Based on the above solutions, the valid numbers that satisfy all the given inequalities are -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. These numbers can be represented on a number line graph by marking the appropriate points and shading the corresponding regions.

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The type of the function is a cube function

The equation of the parabola is y = -1/18(x - 4)² - 1

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

y = (x + 1)³ - 2

The above function has a degree of 3

This means that the type of the function is a cube function

To translate the function from the parent function, we have

Translate left by 1 unit and translate down by 2 units

Also, the function has no minimum or maximum

Here, we have

Vertex = (4, -1)

Point (-2, -3)

A parabola is represented as

y = a(x - h)² + k

Using the vertex, we have

y = a(x - 4)² - 1

Using the point, we have

a(-2 - 4)² - 1 = -3

This gives

a(-2 - 4)² = -2

So, we have

36a = -2

Evaluate

a = -1/18

Recall that

y = a(x - 4)² - 1

So, we have

y = -1/18(x - 4)² - 1

Hence, the equation is y = -1/18(x - 4)² - 1

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Avoidable costs are costs that can be eliminated if the activity that caused them is discontinued. The examples are listed.

Direct materials: Direct materials are the materials that go into a product and can be easily traced to it. For example, the wood used to make a table is a direct material. If a company decides to stop making tables, it can avoid the cost of buying wood.

Direct labor: Direct labor is the labor that is directly involved in making a product. For example, the wages paid to the workers who assemble a car are direct labor. If a company decides to stop making cars, it can avoid the cost of paying direct labor.

Variable overhead: Variable overhead is the overhead costs that vary with the number of units produced. For example, the cost of electricity used to power a factory is a variable overhead cost. If a company decides to stop producing a product, it can avoid the variable overhead costs associated with that product.

Sunk costs: Sunk costs are costs that have already been incurred and cannot be recovered. For example, the cost of research and development for a new product is a sunk cost. If the company decides not to launch the product, it cannot recover the cost of research and development.

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The probability is 5/8 that the second number chosen will exceed the first number by a distance greater than 1/4 unit on the number line when two numbers are randomly chosen between 0 and 1.

To determine the probability that the second number chosen will exceed the first number by a distance greater than 1/4 unit on the number line, we need to consider the possible range of values for both numbers.

Since the two numbers are chosen randomly between 0 and 1, we can visualize this as a unit interval on the number line, where 0 represents the left endpoint and 1 represents the right endpoint.

Let's analyze the scenario where the first number is chosen and labeled as x. The probability that the second number chosen will exceed x by a distance greater than 1/4 unit can be represented by the shaded area on the number line.

To calculate this probability, we need to determine the length of the interval where the second number can be chosen, given that it exceeds x by more than 1/4 unit.

If the first number, x, is chosen between 0 and 3/4 (i.e., x < 3/4), the second number can be chosen in the range (x + 1/4, 1]. The length of this interval is 1 - (x + 1/4) = 3/4 - x.

If the first number, x, is chosen between 3/4 and 1 (i.e., 3/4 ≤ x < 1), the second number can be chosen in the range (3/4, 1]. The length of this interval is 1 - 3/4 = 1/4.

Since the probabilities of choosing a number within each interval are equally likely, we need to calculate the weighted average of these probabilities based on the lengths of the intervals.

The probability of choosing a number between 0 and 3/4 is (3/4 - 0) = 3/4, and the probability of choosing a number between 3/4 and 1 is (1/4 - 0) = 1/4.

Therefore, the overall probability is calculated as:

P = (3/4) * (3/4) + (1/4) * (1/4) = 9/16 + 1/16 = 10/16 = 5/8.

So, the probability that the second number chosen will exceed the first number by a distance greater than 1/4 unit on the number line is 5/8.

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Therefore, for a double batch of 2,000 lbs of flour in mixer 1, you would need 8,000 lbs of water.

To determine the amount of water needed for a double batch of 2,000 pounds (lbs) of flour in mixer 1, we need to consider the ratio of water to flour.

Typically, when making a batch, there is a specific ratio of water to flour. Let's assume the ratio is 1:2, meaning for every 1 pound of flour, we need 2 pounds of water.

For a double batch, we need to multiply the amount of flour by 2:

2,000 lbs of flour × 2 = 4,000 lbs of flour.

Using the ratio, we multiply the amount of flour by the ratio:

4,000 lbs of flour × 2 lbs of water per 1 lb of flour = 8,000 lbs of water.

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The instantaneous rate of change of f(x)=4-2 [tex]x^{2}[/tex] at x=0.5 is −2. This means that at that specific point, the function is changing at a rate of -2 units per unit change in x.

The instantaneous rate of change of a function at a specific point can be found by taking the derivative of the function and evaluating it at that point. In this case, we have the function f(x)=4-2 [tex]x^{2}[/tex] and we want to find the instantaneous rate of change at x=0.5.

To find the derivative of f(x), we apply the power rule for differentiation. Taking the derivative of each term, the derivative of 4 is 0, and the derivative of −2[tex]x^{2}[/tex] is −4x. Therefore, the derivative of f(x) is  (x)=−4x.

Now, to find the instantaneous rate of change at x=0.5, we substitute

x=0.5 into the derivative function. f′(0.5)=−4(0.5)=−2.

This means that the function is changing at a rate of -2 units per unit change in x.

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The vertex (25, 2400) of the parabola reveals the optimal ticket price (25 dollars) that maximizes the profit (2400 dollars) for the theater during the comedy show.

To find the vertex of the parabola, we can use the formula:

x = -b / (2a)

In this case, the function is p(d) = -4d² + 200d - 100, which can be rewritten in the form of ax² + bx + c.

Comparing it with the standard form ax² + bx + c, we can see that a = -4, b = 200, and c = -100.

Now, let's substitute these values into the formula to find the vertex:

d = -200 / (2 * -4)

d = -200 / -8

d = 25

The x-coordinate of the vertex is 25. To find the corresponding y-coordinate, we substitute this value back into the original function:

p(25) = -4(25)² + 200(25) - 100

p(25) = -4(625) + 5000 - 100

p(25) = -2500 + 5000 - 100

p(25) = 2400

The y-coordinate of the vertex is 2400.

Therefore, the vertex of the parabola is (25, 2400).

The vertex reveals important information about the situation. In this case, it represents the optimal point for maximizing profit. The x-coordinate (25) represents the price at which the theater should set the tickets to maximize their profit. The y-coordinate (2400) represents the maximum profit achievable at that price.

Additionally, since the coefficient of the quadratic term (a) is negative (-4), it indicates that the parabola opens downwards, forming a concave shape. This means that as the ticket price increases or decreases from the optimal price, the profit will decrease. Therefore, setting the tickets at a price other than the one corresponding to the vertex would result in a lower profit for the theater.

In summary, the vertex (25, 2400) of the parabola reveals the optimal ticket price (25 dollars) that maximizes the profit (2400 dollars) for the theater during the comedy show.

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Answer:

x = 2

Step-by-step explanation:

P = [tex]\frac{x-2}{x+1}[/tex]

P will equal zero when the numerator is equal to zero , that is

x - 2 = 0 ( add 2 to both sides )

x = 2

P = 0 when x = 2

The situation in which you would add 9 is c) 9 more than a number.

Addition is one of the four basic mathematical operations, including subtraction, multiplication, and division.

Mathematical operations involve the use of mathematical operands on variables, numbers, constants, and values to create algebraic expressions, equations, and functions.

Various Cases:

a) decrease of 9: What is required here is the subtraction of 9 and not addition.

b) descend 9 feet: When an object or person descends 9 feet, the measurement calls for a subtraction operation.

c) 9 more than a number: Addition operation is required to make a number to become 9 more than it was previously.

d) product of a number and 9: The product of a number and 9 involves multiplication and will not make add 9 to the number.

Thus, the addition of 9 is required in case c).

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a) decrease of 9

b) descend 9 feet

c) 9 more than a number

d) product of a number and 9

The interval to the solution is (a) the interval from -7 to 6

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

log(x + 9) + log(x - 9) = 0.47712

Apply the rule of logarithm

So, we have

log(x² - 9) = 0.47712

Take the exponent of both sides

x² - 9 = [tex]10^{0.47712[/tex]

So, we have

x² = 9 + [tex]10^{0.47712[/tex]

Evaluate

x² ≈ 12

Take the square root of both sides

x ≈ ±3.46

This value is between the interval -7 to 6

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Answer:

Yes, the triangle are similar.

Step-by-step explanation:

To prove that 2 triangle are similar you only need to prove that 2 corresponding angles are equal.

The sum of the interior angles add to 180.

< m on the left measures 30 degrees.

80 - 90 - 60 = 30

> t on the right measures 60 degrees.

180 - 90 - 30 = 60

If two corresponding angles of two triangles are equal that forces the third pair to be congruent, since the total of the angles must add up to 180.

Although we are not given the corresponding sides, there are proportional because the angles are equal.

Helping in the name of Jesus.

Answer:

Similar triangles are triangles that have the same shape but different sizes. In other words, if two triangles are similar, then their corresponding angles are congruent and their corresponding sides are in equal proportion.

For Question:

In Δ MSK and ΔQRT

∡S=∡R right angle

∡K=∡T=180°-90°-30°=60° Given

∡M=∡Q=180°-90°-60°=30° Given

Therefore,

Δ MSK  [tex]\bold{\sim}[/tex]  ΔQRT

By AA similarity.

Hence Proved:

Answer:

1/216

Step-by-step explanation:

6×6×6=216

negative exponents meaning reciprocal

so 216/1 means 1/216

Answer:

Hey, math brainiac, check this  out. If our hybrid electric car gets 36 miles per charge and goes through 6 gallons of gas, that means she'll cover another 210 miles. Total trip length? Some sweet 246 miles, bro. That's enough juice to cruise cross-country without ever needin' to stop at a gas station. Talk about efficiency.

The distance traveled by the car by 6 gallons is 246 miles.

Given data:

To determine the distance the car can travel using 6 gallons of gasoline, substitute the value of g = 6 into the formula D = 36 + 35g and evaluate it.

Formula: D = 36 + 35g

Gasoline amount: g = 6

Substituting g = 6 into the formula, we have:

D = 36 + 35(6)

On simplifying the equation:

D = 36 + 210

D = 246

Therefore, the value of D = 246 miles

Hence, the car can travel a distance of 246 miles using 6 gallons of gasoline.

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When r is less than 5 and we substitute r = 4 into the expression [3r-15], the result is -3.

The expression [3r-15] represents an algebraic expression that depends on the value of r. The condition given is that r is less than 5. To evaluate this expression, we substitute the value of r into the expression and simplify it.

Given that r is less than 5, let's substitute r = 4 into the expression:

[3(4) - 15]

= [12 - 15]

= -3

Therefore, when r is less than 5 and we substitute r = 4 into the expression [3r-15], the result is -3.

It's important to note that this answer is specific to the given condition that r is less than 5. If the condition changes or if r is greater than or equal to 5, the result of the expression may be different.

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The domain cannot be (–∞, ∞) as it includes values that are not valid inputs for the inverse cosine function.

The domain of the inverse cosine function, y = cos^(-1)(x) or y = arccos(x), is the set of values for which the function is defined.

In this case, the range of the cosine function, which is the set of values that x can take, is [-1, 1]. Therefore, the domain of the inverse cosine function is the range of the cosine function.

Hence, the correct answer is [–1, 1]. This is because the inverse cosine function is only defined for values of x that fall within the range of the cosine function, which is from -1 to 1.

To clarify further, the inverse cosine function is defined as the inverse of the restricted cosine function. The restricted cosine function is defined on the interval [0, π], which means that its range is [–1, 1]. The inverse cosine function "undoes" the cosine function and maps values from [-1, 1] back to the corresponding input values in [0, π]. Therefore, the domain of the inverse cosine function is [-1, 1].

The options [0, π] and (–∞, ∞) are not correct because they do not correspond to the domain of the inverse cosine function. The range of the inverse cosine function is [0, π], and the inverse cosine function is not defined for values outside the range of the cosine function. Therefore, the domain cannot be (–∞, ∞) as it includes values that are not valid inputs for the inverse cosine function.

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