Write 987.6 in scientific notation.

Answer:

[tex]\dfrac{\ln | \sec (x-b)- \ln | \sec (x-a)}{\sin (a-b)}+\text{C}[/tex]

Step-by-step explanation:

Fundamental Theorem of Calculus

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

[tex]\displaystyle \int \sec(x-a) \sec (x-b)\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{3.5 cm}\underline{Trigonometric Identity}\\\\$\sec \theta=\dfrac{1}{\cos \theta}$\\\end{minipage}}[/tex]

Therefore:

[tex]\implies \displaystyle \int \dfrac{1}{\cos(x-a)} \cdot \dfrac{1}{\cos (x-b)}\:\text{d}x[/tex]

[tex]\implies \displaystyle \int \dfrac{1}{\cos(x-a)\cos (x-b)} \:\text{d}x[/tex]

[tex]\textsf{Multiply the integral by }\dfrac{\sin (a-b)}{\sin (a-b)}:[/tex]

[tex]\implies \displaystyle \int \dfrac{1}{\cos(x-a)\cos (x-b)} \cdot \dfrac{\sin (a-b)}{\sin (a-b)}\:\text{d}x[/tex]

Take the constant outside the integral:

[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin (a-b)}{\cos(x-a)\cos (x-b)} \:\text{d}x[/tex]

Rewrite the numerator:

[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin [(x-b)-(x-a)]}{\cos(x-a)\cos (x-b)} \:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{6 cm}\underline{Trigonometric Identity}\\\\$\sin (A \pm B)=\sin A \cos B \pm \cos A \sin B$\\\end{minipage}}[/tex]

Therefore:

[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin(x-b) \cos (x-a)-\cos (x-b) \sin (x-a)}{\cos(x-a)\cos (x-b)} \:\text{d}x[/tex]

[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin(x-b) \cos (x-a)}{\cos(x-a)\cos (x-b)} -\dfrac{\cos (x-b) \sin (x-a)}{{\cos(x-a)\cos (x-b)}}\:\text{d}x[/tex]

[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \dfrac{\sin(x-b)}{\cos (x-b)} -\dfrac{\sin (x-a)}{\cos(x-a)}\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{3.5 cm}\underline{Trigonometric Identity}\\\\$\tan \theta=\dfrac{\sin \theta}{\cos \theta}$\\\end{minipage}}[/tex]

Therefore:

[tex]\implies \displaystyle \dfrac{1}{\sin (a-b)}\int \tan(x-b)-\tan(x-a)\:\text{d}x[/tex]

[tex]\boxed{\begin{minipage}{4.3 cm}\underline{Integrating $\tan x$}\\\\$\displaystyle \int \tan x\:\text{d}x=\ln | \sec x|+\text{C}$\end{minipage}}[/tex]

Therefore:

[tex]\implies \dfrac{1}{\sin (a-b)}\left[ \ln | \sec (x-b)- \ln | \sec (x-a)\right]+\text{C}[/tex]

[tex]\implies \dfrac{\ln | \sec (x-b)- \ln | \sec (x-a)}{\sin (a-b)}+\text{C}[/tex]

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The given function is exponential growth and the y-intercept is y=11.

Given that the given function is y=11(0.75)ˣ.

Exponential growth or decay describes the process of reducing or growing an amount by a consistent percentage rate over a period of time and it is represented by the function y=a(1-b)ˣ where, y is the final amount, a is the original amount and b is the growth or decay factor and x is the amount of time passed.

If a is positive and b is greater than 1 then it is exponential growth.

If a is positive and b is less than 1 but greater than 0 then it exponential decay.

The graph of the given function y=11(0.75)ˣ is shown in attached image.

From the graph, it is visible that the given function is an exponential growth because a is 11 and it is positive and it is greater than 1.

Now, we will find the y-intercept by substituting x=0, we get

y=11(0.75)⁰

y=11(1)

y=1

Hence, the given function or situation y=11(0.75)ˣ as an example of exponential growth and the y-intercept is 11.

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The inequality used to determine Jaya's maximum number of Gigabytes available for use, within her budget is 60.50 + 5x ≤ 75.

The relation between two expressions that are not equal, employing a sign such as ≠ ‘not equal to’, > ‘greater than’, or < ‘less than’.

Given:

Jaya pays a flat cost of $60.50 per month and $5 per gigabyte and  keep her bill under $75 per month.

According to given question we have

Let the cost of  gigabyte be x

flat cost =$60.50 per month

Budget=  $75 per month

i.e

60.50 + 5x ≤ 75

Therefore, the inequality used to determine Jaya's maximum number of Gigabytes available for use, within her budget is 60.50 + 5x ≤ 75.

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The result of division of (x² - x - 12) by (x - 4) is (x + 3).

The given equation is;

(x² - x - 12) / (x - 4)

where, (x² - x - 12) is the numerator and (x - 4) is the denominator.

Find the factors of the numerator.

= (x - 4)(x + 3)/(x - 4)

Cancel the values (x - 4) from the numerator an denominator part.

= (x + 3)

Thus, the solution of the factorization is found as (x + 3).

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Answer:[tex]\frac{-11x^2+135}{5}[/tex]

Step-by-step explanation:

Answer:The answer would be negative.

Step-by-step explanation:There is already a negative balance of -$25 and you would add $23.-25+23=-2.So therefore the answer would be negative.

Please give brainliest if answer is correct.

The temperature outside at night is 2 degrees.

One degree Celsius, for instance, is one-hundredth of the temperature difference between the point at which water changes state from solid to liquid to its gaseous stage. A degree can be defined as a fixed change in temperature measured against a specific scale.

Let's say the temperature outside is x.

Now, the initial temperature was - 17 degrees, then it rises to 9 degrees during the day before it drops 10 degrees at night.

So,

x = - 17 degrees

The temperature rises by 9 degrees.

Then,

x = - 17 degrees + 9 degrees

x = - 8 degrees

The temperature drops 10 degrees at night.

Then,

x = -8 degrees + 10 degrees

x = 2 degrees

Hence, the temperature at night is 2 degrees.

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log (x + 4) = 8 => x =  [tex]10^8[/tex], by properties of logarithm.

Given:

log ( x + 4) = 8

Raising both sides by 10, in order to remove the logarithm.

=> [tex]10^{log_{10}(x+4)} = 10^{8}[/tex]

=> x + 4 = [tex]10^8[/tex] (as [tex]a^{log_a(x)} = x[/tex] )

=> x =  [tex]10^8[/tex] (as the value of  [tex]10^8[/tex] is so large, subtracting 4 from it won't make much difference).

Hence, log (x + 4) = 8 => x =  [tex]10^8[/tex], by properties of logarithm.

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Answer:

4

Step-by-step explanation:

15-2x=7

8=2x

4=x

---

HOPE THIS HELPS <3

Answer:

x=11

Step-by-step explanation:

let us name the unknown number: x

15 - 2x = 7

collect like terms,

-2x= 7 - 15

-2x= - 8

divide both sides by -2( coefficient of x)

-2x = -8

__ __

-2 -2

x= 4

The length of the side of the equilateral triangle is 20.76 inches.

Here it is given that the length of the altitude of the equilateral triangle is 18 inches.

The formula of the altitude(h) of the equilateral triangle is

h = [tex]\frac{\sqrt{3} }{2}[/tex]a

Here a is the side of an equilateral triangle.

So we have

18 = [tex]\frac{\sqrt{3} }{2}[/tex]a

a = 18 × 2/[tex]\sqrt{3}[/tex]

For the rationalization of the fraction multiply numerator and denominator by [tex]\sqrt{3}[/tex].

a = [tex]\frac{18 * 2 * \sqrt{3} }{\sqrt{3}\sqrt{3} }[/tex]

  = [tex]\frac{18 * 2 * \sqrt{3} }{3}[/tex]

  = 6 × 2 × √3

  = 12×√3

  = 12 × 1.73

  = 20.76 inches

Therefore the length of the side of the triangle is 20.76 inches.

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The total amount that's is borrowed by Benjamin is $3.75.

It should be noted that from the information, Benjamin borrowed $0.75 from his friend cltyde five days last week to buy ice cream.

Therefore, the product will be:

= Amount borrowed × Number of times

= $0.75 × 5

= $3.75

The amount borrowed is $3.75.

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By applying direct proportion, the information should be filled in the table are as follows:

Number of containers                     Weight (in pounds)

4                                                               16,000

8                                                               32,000

10                                                              40,000

A proportion can be defined as an equation which is typically used to represent (indicate) the equality of two (2) ratios. This ultimately implies that, proportions can be used to establish that two (2) ratios are equivalent and solve for all unknown quantities.

Mathematically, a direct proportion can be represented the following equation:

y = kx

Where:

Since the boat carried containers that weigh 4000 pounds each, we would multiply each of containers by 4000 as follows:

Number of containers                     Weight (in pounds)

4                                                               16,000

8                                                               32,000

10                                                              40,000

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Answer:

h = 0.01g

Step-by-step explanation:

[tex]{ \tt{g = kh + c}}[/tex]

When g is 0, h is 0 hence c is 0

[tex]{ \tt{g = kh}}[/tex]

when g is 1, h is 0.01

[tex]{ \tt{1 = 0.01k}} \\ { \tt{k = 100}} \\ { \boxed{ \tt{ equation \to \: {g = 100h}}}} \\ { \rm{or}} \\ { \boxed{ \tt{equation \to \: h = 0.01g}}}[/tex]

Answer: 11 and 26

Step-by-step explanation:

Answer:

mmmm

Step-by-step explanation:

!!!

Answer:

see explanation

Step-by-step explanation:

∠ CDF and 137° are a linear pair and sum to 180° , then

∠ CDF = 180° - 137° = 43°

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

the sum of the 3 angles in Δ CDF = 180° , then

∠ CFD + 43° + 32° = 180°

∠ CFD + 75° = 180° ( subtract 75° from both sides )

∠ CFD = 105°

then

∠ AFB = ∠ CFD = 105° ( vertically opposite angles are congruent )

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

∠ ABF= ∠ FCD = 32° ( alternate angles are congruent )

Answer:

2456x =

Step-by-step explanation:

Simple i know but you didnt really specify.

Answer: 4,912/2=2,456

Step-by-step explanation:hope it helps, brainliest? i could really use it

Option number (4) i.e., (3+7)+2=3+(7+2)  illustrates the associative property of addition.

As per the question statement, we are supposed to identify the equation which illustrates the associative property of addition.

Before solving this, we should be aware about the concept of associative property of addition which states that rearranging the parentheses in an expression will not change the result i.e., a+(b+c) = (a+b)+c

Therefore using the above property, we conclude that option number (4) i.e., (3+7)+2=3+(7+2)  illustrates the associative property of addition.

As (3+7)+2=3+(7+2) can be written in the standard form a+(b+c)=(a+b)+c which is  a characteristic equation of associative property of addition.

      e.g.,  a+(b+c)=(a+b)+c OR a*(b*c)=(a*b)*c

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Answer:

Solution:  x < -2 or x > 3

Interval notation:  (-∞, -2) ∪ (3, ∞)

Step-by-step explanation:

Given inequality:

[tex]||2x-1|-2| > 3[/tex]

Apply the absolute rule:

[tex]\textsf{If $|u| > a$, $a > 0$ \;then \;$u > a$ \;or \;$u < -a$}.[/tex]

Therefore:

[tex]\textsf{Case 1}: \quad |2x-1|-2 > 3[/tex]

[tex]\textsf{Case 2}: \quad |2x-1|-2 < -3[/tex]

Solve each case independently.

Isolate the absolute value on one side of the equation:

[tex]\begin{aligned}\implies |2x-1|-2& > 3\\\implies |2x-1| & > 5 \end{aligned}[/tex]

Apply the absolute rule:

[tex]\textsf{If $|u| > a$, $a > 0$ \;then \;$u > a$ \;or \;$u < -a$}.[/tex]

[tex]\begin{aligned}\underline{\textsf{Equation 1}} & & \quad \quad\underline{\textsf{Equation 2}}\\2x-1 & > 5 & 2x-1 & < -5\\2x& > 6 & 2x& < -4\\x& > 3 & x& < -2\end{aligned}[/tex]

Therefore, x < -2 or x > 3.

Isolate the absolute value on one side of the equation:

[tex]\begin{aligned}\implies |2x-1|-2& < -3\\\implies |2x-1| & < -1 \end{aligned}[/tex]

As an absolute value cannot be less than zero, there is no solution for x∈R.

Solution:  x < -2 or x > 3

Interval notation:  (-∞, -2) ∪ (3, ∞)

For given survey of 62 customers was taken at a bookstore regarding the types of books purchased,

a) The number of customers surveyed who purchased only science fiction = 9

b) The number of customers surveyed who purchased mysteries and science fiction, but not romance novels = 13

c) The number of customers surveyed who purchased mysteries or science fiction = 51

d) The number of customers surveyed who purchased mysteries or science fiction, but not romance novels = 46

e) The number of customers surveyed who purchased exactly two types of books = 2

In this question,

we have been given a survey of 62 customers was taken at a bookstore regarding the types of books purchased.

38 customers purchased mysteries,

31 purchased science fiction,

21 purchased romance novels,

18 purchased mysteries and science fiction,

12 purchased mysteries and romance novels,

9 purchased science fiction and romance novels,

and 5 purchased all three types of books.

Let S represents the set of customers who purchased science fiction books,

R represents the set of customers who purchased romance novels books and

M represents the set of customers who purchased mysteries books.

From given information,

n(S) = 31

n(R) = 21

n(M) = 38

n(M ∩ S) = 18

n(M ∩ R) = 12

n(S ∩ R) = 9

n(M ∩ R ∩ S) = 5

a) we need to find the number of customers surveyed who purchased only science fiction.

= n(S) - n(M ∩ S) - n(S ∩ R) - n(M ∩ R ∩ S)

= 31 - (18 - 5) - (9 - 5) - 5

= 31 - 13 - 4 - 5

= 9

b)  we need to find the number of customers surveyed who purchased mysteries and science fiction, but not romance novels

= n(M ∩ S) - n(M ∩ R ∩ S)

= 18 - 5

= 13

c) we need to find the number of customers surveyed who purchased mysteries or science fiction

n(M ∪ S) = n(M) + n(S) - n(M ∩ S)

              = 38 + 31 - 18

              = 51

d) we need to find the number of customers surveyed who purchased mysteries or science fiction, but not romance novels

= n(M ∪ S) - n(M ∩ R ∩ S)

= 51 - 5

= 46

e)  we need to find the number of customers surveyed who purchased exactly two types of books

=  n(M ∩ S) - n(S ∩ R) - n(M ∩ R)

= (18 - 5) - (9 - 5) - (12 - 5)

= 13 - 4 - 7

= 2

Therefore, for given survey of 62 customers was taken at a bookstore regarding the types of books purchased,

a) The number of customers surveyed who purchased only science fiction = 9

b) The number of customers surveyed who purchased mysteries and science fiction, but not romance novels = 13

c) The number of customers surveyed who purchased mysteries or science fiction = 51

d) The number of customers surveyed who purchased mysteries or science fiction, but not romance novels = 46

e) The number of customers surveyed who purchased exactly two types of books = 2

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Based on the dimensions of the shape, the length of PR can be found to be 19.8 units.

The given triangle is an equilateral triangle which means that all the sides are equal.

PR = QR

This means that:

2n + 9 = 7n - 18

Solving for n gives:

7n - 2n = 18 + 9

5n = 27

n = 5.4

The length of PR is therefore:

= 2n + 9

= 2 (5.4) + 9

= 19.8 units

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|x - 50| = 14 is the equation used to find the locations of the outer edges of the stage.

Absolute value function

The absolute function is defined by

 |x| = x, x≥0

 |x| = -x, x<0

It measures the distance of a point x to the origin, let's take some example

|-8| = 8

|-5| = 5

Here it is given that the center of the stage is on the 50-yard line, and the distance from the center to the outer edges is 14 yards.

So from this, we get an equation,

|x - 35| = 11

So the solution is given as follows:

First edge:

x - 50 = -14

x = 36 yards line

Second edge:

x- 50 = 14

x = 64 yard line

Therefore the equation used to find the location of the outer edges of the stage is |x - 50| = 14.

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If a student population of 1200 was estimated to increase by 15% in the next five years but the population actually increased by 20%, then the most estimated and actual student populations are equal to 1380 and 1440 students respectively, and the percent error is equal to 4.2%.

The estimated and actual student populations can be calculated as follows,

Estimated increase = 15% = 15 / 100 = 0.15

Initial population = 1200

0.15 × 1200 = 180

Estimated population in the next five years = 1200 + 180 = 1380

Actual increase = 20% = 20 / 100 = 0.2

Initial population = 1200

0.2 × 1200 = 240

Actual population in the next five years = 1200 + 240 = 1440

We can find the percentage error as follows,

percentage error = [(actual value - estimated value) / actual value] × 100

percentage error = [(1440 - 1380) / 1440] × 100

percentage error = [60 / 1440] × 100

percentage error = 0.042 × 100 = 4.2%

Hence the estimated and actual student populations is calculated to be 1380 and 1440 respectively and the percentage error is equal to 4.2%.

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Answer:

if any number power of 0 is 1

so, (4)^0 =1

if any number power of -1 is equal to one divide this number.

like;

a^(-1) = 1/a

Therefore,

2^(-3) = 2^(-1) x 2^(-1) x 2^(-1)

= 1/2 x 1/2 x 1/2

Hence solution of 2^(-3). (4) ^0 is,

1/2 x 1/2 x 1/2 x 1

After solving, the factors of equation 3x² = 18x-24 are:

So,

Given equation: 3x² = 18x-24

Then,

Factors are: (x-4) and (3x-6)

Therefore, after solving, the factors of equation 3x² = 18x-24 is:

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Answer:

The slope of the line is 12

(Sorry for the bad quality image)

Explanation:

To find the average rate of change, calculate the change in y over the change in x.

m = 12

Answer:

Rate of change = 12
y-intercept = (0, 20)

Step-by-step explanation:

Part 1) Rate of Change
The rate of change (or slope) is found using the formula [tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex].
To use this formula, we must take the coordinates of two of the given points in the table and substitute them. For simplicity, I'll take the points (1, 32) and (4, 68).

[tex]m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{68 - 32}{4 - 1} = \frac{36}3 = 12[/tex]

Therefore, the rate of change is 12.

Part 2) y-intercept
To find the y-intercept, we must find the value of the function where [tex]x = 0[/tex]. Since we have both the value of the function where [tex]x = 1[/tex] and we know that the rate of change is 12, we can simply subtract the rate of change from the y-value of the function at the point where [tex]x = 1[/tex].

The y-value of the function where [tex]x = 1[/tex] is 32, therefore, the y-value of the function at [tex]x = 0[/tex] is [tex]32 - 12 = 20[/tex]

The y-intercept is (0, 20).

Answer: The length of First piece = 6

The length of second piece = 18

The length of third piece = 30  

Step-by-step explanation:

Given data,

A 54-inch board is to be cut into three pieces.

so that the second piece is 3 times as long as the first piece and the third piece is 5 times as long as the first piece.

So, we can write,

Let us assume, first piece is represented by = x

Then,

second piece is 3 times as long as the first piece

So, we can write,

second piece is represented by = 3 ( first piece )

second piece is represented by = 3x

Then,

third piece is 5 times as long as the first piece

So, we can write,

third piece is represented by = 5 ( first piece )

third piece is represented by = 5x

So, we can find the all 3 pieces length,

we can solve it :
combine all three pieces = x + 3x + 5x

length of all three pieces = x + 8x

                                          = 9x

 Total board is to be cut into three pieces = 54

Hence,

                         9 x = 54

                          x = 54/9

                          x = 6

Now we know that the base length, x, is equal to 6.

From there, we can find the length of the all three pieces is :

The length of First piece = x = 6

The length of second piece = 3x

                                         = 3(6)

                                         = 18

The length of third piece = 5x

                                         = 5(6)

                                         = 30

Therefore,

The length of First piece = 6

The length of second piece = 18

The length of third piece = 30  

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Carl made a mistake by multiplying the two values in the quotient rather than dividing them.

So, the result of 4 divided by 2 1/3 and an explanation of Carl's error:

Thus, if we divide 4 2/3 by 2 1/3, we get:

Carl made the error of assuming his quotient (2) was equal to the value of the terms it contained.

Therefore, Carl made a mistake by multiplying the two values in the quotient rather than dividing them.

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Math reality is more restrictive and logical, physical reality is more "free" and well-behaved.

Mathematics are a logical construct, thus, everything in the "mathematical realty" must follow a certain logic.

For example, in math, always that you do a simplification (like rounding, applying a theorem, using an integration property, etc) you need to prove logically why you can do that.

While on physics we assume the reality is "nice" and we can always apply the simplifications. This is because most of the functions that represent physics are nice (continuous, differentiable, etc)  functions, in the same way, most of the matrices are square matrices, and so on.

Concluding, for example in math the number 4.99999 is exactly 4.9999

On physics, if that same number represents a measure, for example:

4.99999 meters is practically equal to 5 meters.

Math reality is more restrictive and logical, physical reality is more "free" and well-behaved.

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If the inverse of f exists, it is represented by f⁻¹ and exists only if f is a bijective function.

The inverse of the function is f⁻¹ = - (x² - 3/2)

Domain is 3/2 to positive infinity

Range is 0 to positive infinity.

The inverse function of a function f is a function that reverses the operation of f. The inverse of f exists if and only if f is bijective, and it is denoted by f⁻¹ if it exists.

A function's inverse is not always a function. To ensure that the inverse function is also a function, the original function must be a one-to-one function. A one-to-one function is one in which each second element corresponds to exactly one first element.

Let the given function be f(x) = √-2x+3

y = √-2x+3

simplifying the value of x, we get

x = √-2x+3

x² = -2y + 3

simplifying the above equation, we get

x² - 3 = -2y

y = - (x² - 3/2)

Therefore, the inverse function be f⁻¹ = - (x² - 3/2)

Domain is 3/2 to positive infinity because a negative number cannot be square-rooted.

Range is 0 to positive infinity.

f⁻¹ = - (x² - 3/2)

The domain of the inverse is negative infinity to positive infinity, which is not a mirror image of the range of the original equation.

Anything squared is a positive number or zero.

The minimum value of the range is

f⁻¹ = - (x² - 3/2)

f⁻¹ = -(-3/2)

f⁻¹ = 3/2

The range of the inverse is 3/2 to positive infinity, a mirror of the domain of the original equation.

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