Find the equations of the asymptotes of the hyperbola defined by the equation shown below. If necessary, round to the nearest tenth. 100pts

the center of the given ellipse is (-1, 1).Hence, the required answer is (-1, 1).

The given equation of the ellipse is

100pts9x²+4y²+18x - 8y-23=0.

To find the center of the ellipse, Rearrange the given equation of the ellipse to standard form by completing the square. To complete the square for x terms, we need to add

(18/2)²=9²=81

to both sides of the equation. To complete the square for y terms, we need to add

(-8/2)²=4²=16

to both sides of the equation.

100pts9x²+18x+4y²-8y=23+81+16-100pts100pts(9x²+18x+81) + 100pts(4y²-8y+16) = 120100pts(3x+3)² + 100pts(2y-2)² = 120 + 100pts100pts3(x+1)² + 100pts2(y-1)² = 180

The standard form of the given equation of the ellipse is:

100pts(3(x+1)²)/180 + (2(y-1)²)/180 = 1

Divide throughout by

180:100pts(3(x+1)²)/180 + (2(y-1)²)/180 = 1 Simplify:100pts(3(x+1)²)/36 + (2(y-1)²)/90 = 1

The center of the ellipse is (-1, 1) (h, k), where h is the x-coordinate of the center and k is the y-coordinate of the center.

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Answer:

Center = (-1, 1)

Step-by-step explanation:

To find the center of the ellipse, we first need to write the equation in its standard form by completing the square.

Given equation:

[tex]9x^2+4y^2+18x-8y-23=0[/tex]

Arrange the equation so that all the terms with variables are on the left side and the constant is on the right side.

[tex]9x^2+18x+4y^2-8y=23[/tex]

Factor out the coefficient of the x² term and the coefficient of the y² term:

[tex]9(x^2+2x)+4(y^2-2y)=23[/tex]

Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:

[tex]9(x^2+2x+1)+4(y^2-2y+1)=23+9(1)+4(1)[/tex]

Factor the two perfect trinomials on the left side and simplify the right side:

[tex]9(x+1)^2+4(y-1)^2=36[/tex]

Divide both sides by 36 so the right side equals 1:

[tex]\dfrac{9(x+1)^2}{36}+\dfrac{4(y-1)^2}{36}=\dfrac{36}{36}[/tex]

[tex]\dfrac{(x+1)^2}{4}+\dfrac{(y-1)^2}{9}=1[/tex]

The standard form of the equation of an ellipse with center (h, k) is:

[tex]\boxed{\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1}[/tex]

Comparing the rewritten equation with the standard equation, we can determine that h = -1 and k = 1.

Therefore, the center (h, k) of the ellipse is (-1, 1).

Step-by-step explanation:

Y = 2 sin 3x            '2' is the amplitude

                      ( 'sin x' usually has amplitude of '1'...then you multiply it by '2' )

                                '3' changes the period

Answer:

Step-by-step explanation:

he amplitude of the sinusoid graph y=2sin3x is 2.

The function for which f(x) is equal to f⁻¹(x) is: C. [tex]f(x) = \frac{x+1}{x-1}[/tex]

In this exercise, you are required to determine the inverse of the function f(x) with an equivalent inverse function f⁻¹(x). This ultimately implies that, we would have to swap (interchange) both the independent value (x-value) and dependent value (y-value) as follows;

[tex]f(x) = y = \frac{x+6}{x-6} \\\\x=\frac{y+6}{y-6}[/tex]

x(y - 6) = y + 6

y = xy - 6x - 6

f⁻¹(x) = (-6x - 6)/(x - 1) ⇒ Not equal.

Option B.

[tex]f(x) = y = \frac{x+2}{x-2} \\\\x=\frac{y+2}{y-2}[/tex]

x(y - 2) = y + 2

y = xy - 2x - 2

f⁻¹(x) = (-2x - 2)/(x - 1) ⇒ Not equal.

Option C.

[tex]f(x) = y = \frac{x+1}{x-1} \\\\x=\frac{y+1}{y-1}[/tex]

x(y - 1) = y + 1

y - xy = x + 1

f⁻¹(x) = (x + 1)/(x - 1) ⇒ equal.

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An estimate for the mean is 47.6 kg.

In Mathematics and Geometry, the mean for this set of data can be calculated by using the following formula:

Mean = [F(x)]/n

Cumulative frequency = 10 + 7 + 2 + 8 + 3

Cumulative frequency = 30

For the total number of data based on the frequency, we have;

Total weight, F(x) = 10(40) + 7(52.5) + 2(65) + 8(77.5) + 3(90)

Total weight, F(x) = 40 + 367.5 + 130 + 620 + 270

Total weight, F(x) = 1427.5

Now, we can calculate the mean weight as follows;

Mean = 1427.5/30

Mean = 47.6 kg.

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Answer:

W = V/(LH)

Step-by-step explanation:

All we are doing is isolating W. Since V=LWH, then dividing both sides by LH will put W by itself on the right-hand side, you have V/(LH) = W as your equation

Answer:

  f(3) = 1.432

Step-by-step explanation:

You want to know the third term of the sequence defined by ...

The terms of the sequence can be found one at a time by evaluating the recursive relation. The attached calculator output shows the first three terms are ...

  f(1) = 0.2 . . . . . . . given

  f(2) = 0.4(0.2) +1 = 1.08

  f(3) = 0.4(1.08) +1 = 1.432

The third term of the sequence is 1.432.

__

Additional comment

The explicit form of the function is ...

  f(n) = 5/3 -11/3(2/5)^n

Terms will asymptotically approach a value of 5/3.

<95141404393>

Answer:

(-4,4)

Step-by-step explanation:

You rewrite the terms:

(x + 4)^2 => [x - (-4)]^2

(y - 4)^2 => [y - (4)]^2

so h = -4 and k = 4

so center of ellipse is (h,k) or (-4,4)

Answer:

Center = (-4, 4)

Step-by-step explanation:

The standard form of the equation of an ellipse with center (h, k) is:

[tex]\boxed{\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1}[/tex]

The given equation is:

[tex]\dfrac{(x+4)^2}{25}+\dfrac{(y-4)^2}{9}=1[/tex]

Comparing the given equation with the standard form, we can see that h = -4 and k = 4. Therefore, the center (h, k) of the ellipse is (-4, 4).

Lim as x approaches 0: (sin3x)/(2x-Sinx)

Taking the derivative of the numerator and denominator separately:

Numerator: d/dx(sin3x) = 3cos3x

Denominator: d/dx(2x - sinx) = 2 - cosx

Now, evaluate the limit using L'Hôpital's Rule:

Lim as x approaches 0: (3cos3x)/(2 - cosx)

Plugging in x = 0:

Lim as x approaches 0: (3cos(0))/(2 - cos(0))

= 3/2

Therefore, the limit as x approaches 0 of (sin3x)/(2x-Sinx) is 3/2.

Lim as x approaches infinity: x^-1 lnx

Taking the derivative of the numerator and denominator separately:

Numerator: d/dx(x^-1 lnx) = (1/x)lnx

Denominator: d/dx(1) = 0

Since the denominator is 0, we cannot apply L'Hôpital's Rule. However, we can still evaluate the limit:

Lim as x approaches infinity: x^-1 lnx

As x approaches infinity, the natural logarithm (lnx) grows without bound, so the overall limit is 0.

Therefore, the limit as x approaches infinity of x^-1 lnx is 0.

Lim x approaches infinity: x/ e^x

Taking the derivative of the numerator and denominator separately:

Numerator: d/dx(x) = 1

Denominator: d/dx(e^x) = e^x

Now, evaluate the limit using L'Hôpital's Rule:

Lim as x approaches infinity: 1/ e^x

As x approaches infinity, the exponential function e^x grows without bound, so the overall limit is 0.

Therefore, the limit as x approaches infinity of x/ e^x is 0.

Answer:

Step-by-step explanation:

step 1:

determining the values for standard form for the equation of a line,

y = mx + c

Step 2:

calculation of m, where m is the gradient or slope which determines how steep the line is.

step 3:

calculation of c, where c is the height at which the line crosses the y - axis also known as y - intercept

a. There are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.

b. There are 14,950 ways to form a 4-person committee from a class of 26 members.

a. To select a president, a vice president, and a secretary from a class of 26 members, we can use the concept of permutations.

For the president position, we have 26 choices. After selecting the president, we have 25 choices remaining for the vice president position. Finally, for the secretary position, we have 24 choices left.

The total number of ways to select a president, a vice president, and a secretary is obtained by multiplying the number of choices for each position:

Number of ways = 26 * 25 * 24 = 15,600

Therefore, there are 15,600 ways to select a president, a vice president, and a secretary from a class of 26 members.

b. To form a 4-person committee from a class of 26 members, we can use the concept of combinations.

The number of ways to choose a committee of 4 people can be calculated using the formula for combinations:

Number of ways = C(n, r) = n! / (r!(n-r)!)

where n is the total number of members (26 in this case) and r is the number of people in the committee (4 in this case).

Plugging in the values, we have:

Number of ways = C(26, 4) = 26! / (4!(26-4)!)

Calculating this expression, we get:

Number of ways = 26! / (4! * 22!)

Using factorials, we simplify further:

Number of ways = (26 * 25 * 24 * 23) / (4 * 3 * 2 * 1) = 14,950

Therefore, there are 14,950 ways to form a 4-person committee from a class of 26 members.

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con un conjunto de deshielo determinante igual a 4, es x = 2.

Para determinar el valor de x en la matriz x 2

5 7

dado que el conjunto de deshielo determinante es igual a 4, necesitamos utilizar la propiedad de que el determinante de una matriz 2x2 se puede calcular utilizando la siguiente formula:

determinante = (a × d) - (b × c)

Donde a, b, c, y d son los elementos de la matriz.

En este caso, tenemos la matriz:

x 2

5 7

Aplicando la formula del determinante, podemos establecer la siguiente ecuacion:

( x × 7 ) - ( 2 × 5 ) = 4

Simplificando la ecuacion, obtenemos:

7x - 10 = 4

A continuacion, vamos a resolver la ecuacion para encontrar el valor de x:

7x = 4 + 10

7x = 14

Dividiendo ambos lados de la ecuacion por 7, obtenemos:

x = 2

Por lo tanto, el valor de x en la matriz x 2

5 7

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Answer:

53)

[tex]f(x) = \frac{7(x - 4)(x + 6)}{(x + 4)(x + 5)} [/tex]

The expression (3√2) / (3√6) with a rationalized denominator is 3√9 / 6. Option C is the correct answer.

To rationalize the denominator in the expression (3√2) / (3√6), we can multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of √6 is -√6, so we multiply the expression by (-√6) / (-√6):

(3√2 / 3√6) * (-√6 / -√6)

This simplifies to:

-3√12 / (-3√36)

Further simplifying, we have:

-3√12 / (-3 * 6)

-3√12 / -18

Finally, we can cancel out the common factor of 3:

- 3√9 / - 6.

Simplifying further, we get:

3√9 / 6.

Option C is the correct answer.

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Answer:

To find the average speed of the truck, we can divide the total distance travelled by the total time taken.

Average speed = Total distance / Total time

In this case, the truck travelled 261 miles in 4 hours.

Average speed = 261 miles / 4 hours

Average speed = 65.25 mph (rounded to two decimal places)

Therefore, the truck was averaging approximately 65 mph on this trip.

The correct option is (a) 65 mph.

The cafeteria can conclude that a majority of the senior high school students like the newly served snack.

To determine if more than 50% of the students like the newly served snack, we need to analyze the responses of the 60 randomly selected students.

Analyzing the responses:

Out of the 60 students surveyed, we have:

- Number of students who responded with "1" (liking the snack): 32 students.

- Number of students who responded with "0" (not liking the snack): 28 students.

To determine the percentage of students who liked the snack, we divide the number of students who liked it by the total number of students surveyed and multiply by 100: (32/60) * 100 = 53.33%.

Since the percentage of students who liked the newly served snack is 53.33%, which is greater than 50%, we can conclude that more than 50% of the students like the snack based on the given survey results.

Therefore, the cafeteria can conclude that a majority of the senior high school students like the newly served snack.

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Answer: -13x² - 4x - 7

Step-by-step explanation:

    We will subtract (9x² + 4x) from (-4x² - 7).

Given:

  -4x² - 7 - (9x² + 4x)

Distribute the negative:

  -4x² - 7 - 9x² - 4x

Reorder terms by degree:

  -4x² - 9x² - 4x - 7

Combine like terms:

  -13x² - 4x - 7

The surface area of the triangular pyramid is approximately 331.93 square yards.

To find the surface area of the triangular pyramid, we need to calculate the areas of its individual components and then sum them up.

The triangular pyramid has a base shaped like an equilateral triangle. The legs of the equilateral triangle are all 11 yards long, and the height of the equilateral triangle is 9.5 yards. The formula to calculate the area of an equilateral triangle is:

Area = (√3/4) * [tex]side^2[/tex]

Plugging in the values, we get:

Area of the base equilateral triangle = (√3/4) * 11^2 ≈ 52.43 square yards

The triangular pyramid also has three triangular faces. Each face is an isosceles triangle, with two sides measuring 11 yards (same as the sides of the base equilateral triangle) and a slant height of 17 yards. We can use the formula for the area of an isosceles triangle:

Area = (1/2) * base * height

Since the base of the isosceles triangle is 11 yards and the height is 17 yards, the area of each triangular face is:

Area of each triangular face = (1/2) * 11 * 17 = 93.5 square yards

Now, we can calculate the total surface area of the triangular pyramid by summing up the areas of the base and the three triangular faces:

Surface area = Area of the base equilateral triangle + 3 * Area of each triangular face

Surface area = 52.43 + 3 * 93.5

Surface area ≈ 331.93 square yards

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Step-by-step explanation:

a = k/m    or       ma = k    

using 4 and 9     4* 9 = k = 36

then the equation becomes:

    ma = 36    

  using a = 6

      6 * m = 36     shows m = 6 kg

Answer:

64 and 65

Step-by-step explanation:

Let the two lockers be x and (x+1) since they are consecutive

The sum is 129 so,

x + (x+1) = 129

2x + 1 = 129

2x = 128

x = 64

x + 1 = 65

The locker numbers are 64 and 65

Third option is correct.The midpoint of WZ of WXYZ with the vertices W(0, 0), X(h, 0), Y(h,b), and Z(0, b) is (0, b/2).

To find the midpoint of segment WZ, we need to average the x-coordinates and the y-coordinates of the endpoints.

The coordinates of point W are (0, 0), and the coordinates of point Z are (0, b).

To find the x-coordinate of the midpoint, we average the x-coordinates of W and Z:

(x-coordinate of W + x-coordinate of Z) / 2 = (0 + 0) / 2 = 0 / 2 = 0

To find the y-coordinate of the midpoint, we average the y-coordinates of W and Z:

(y-coordinate of W + y-coordinate of Z) / 2 = (0 + b) / 2 = b / 2

Therefore, the midpoint of segment WZ is (0, b/2).

So, the correct answer is (0, b/2).

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Answer:

The overall average nickel content of the rock samples is approximately 7.97%.

Step-by-step explanation:

To find the overall average nickel content of the rock samples, we need to take into account the number of samples from each location. Since we know the average nickel content of each set of samples, we can use a weighted average formula:

overall average nickel content = (total nickel content from first location + total nickel content from second location) / (total weight of samples from both locations)

To calculate the total nickel content from each location, we need to multiply the average nickel content by the number of samples:

total nickel content from first location = 8.43% x 1 sample = 8.43%

total nickel content from second location = 7.81% x 6 samples = 46.86%

To calculate the total weight of the samples from both locations, we need to add up the number of samples:

total weight of samples from both locations = 1 + 6 = 7

Now we can substitute these values into the formula and calculate the overall average nickel content:

overall average nickel content = (8.43% + 46.86%) / 7 ≈ 7.97%

Therefore, the overall average nickel content of the rock samples is approximately 7.97%.

1. 1/3 of a full rotation: The coordinates of point P after rotating 1/3 of a full rotation counterclockwise are approximately (0.5, 0.866).

2. 1/2 of a full rotation: The coordinates of point P after rotating 1/2 of a full rotation counterclockwise are (-1, 0).

3. 2/3 of a full rotation: The coordinates of point P after rotating 2/3 of a full rotation counterclockwise are approximately (-0.5, -0.866).

1/3 of a full rotation:

To find the coordinates of point P after rotating 1/3 of a full rotation counter clockwise, we need to determine the angle of rotation.

A full rotation around the unit circle is 360 degrees or 2π radians.

Since 1/3 of a full rotation is (1/3) [tex]\times[/tex] 360 degrees or (1/3) [tex]\times[/tex] 2π radians, we have:

Angle of rotation = (1/3) [tex]\times[/tex] 2π radians

Now, let's use the properties of the unit circle to find the new coordinates.

At the initial position, point P is located at (1, 0).

Rotating counterclockwise by an angle of (1/3) [tex]\times[/tex] 2π radians, we move along the circumference of the unit circle.

The new coordinates of point P after the rotation will be (cos(angle), sin(angle)).

Substituting the angle of rotation into the cosine and sine functions, we get:

New coordinates of P = (cos((1/3) [tex]\times[/tex] 2π), sin((1/3) [tex]\times[/tex] 2π))

Calculating the values:

cos((1/3) [tex]\times[/tex] 2π) ≈ 0.5

sin((1/3) [tex]\times[/tex] 2π) ≈ 0.866

Therefore, the coordinates of point P after rotating 1/3 of a full rotation counterclockwise are approximately (0.5, 0.866).

1/2 of a full rotation:

Following a similar process, when rotating 1/2 of a full rotation counterclockwise, we have an angle of (1/2) [tex]\times[/tex] 2π radians.

New coordinates of P = (cos((1/2) [tex]\times[/tex] 2π), sin((1/2) [tex]\times[/tex] 2π))

Calculating the values:

cos((1/2) [tex]\times[/tex] 2π) = cos(π) = -1

sin((1/2) [tex]\times[/tex] 2π) = sin(π) = 0

Therefore, the coordinates of point P after rotating 1/2 of a full rotation counterclockwise are (-1, 0).

2/3 of a full rotation:

For a rotation of 2/3 of a full rotation counterclockwise, the angle is (2/3) [tex]\times[/tex] 2π radians.

New coordinates of P = (cos((2/3) [tex]\times[/tex] 2π), sin((2/3) [tex]\times[/tex] 2π))

Calculating the values:

cos((2/3) [tex]\times[/tex] 2π) ≈ -0.5

sin((2/3) [tex]\times[/tex] 2π) ≈ -0.866

Therefore, the coordinates of point P after rotating 2/3 of a full rotation counterclockwise are approximately (-0.5, -0.866).

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Answer:

  f(x +h) = 4x² +4h² +8xh +5x +5h +3

  (f(x+h) -f(x))/h = 4h +8x +5

  f'(x) = 8x +5

Step-by-step explanation:

For f(x) = 4x² +5x +3, you want the simplified expression f(x+h), the difference quotient (f(x+h) -f(x))/h, and the value of that at h=0.

Put (x+h) where h is in the function, and simplify:

  f(x+h) = 4(x+h)² +5(x+h) +3

  = 4(x² +2xh +h²) +5x +5h +3

  f(x +h) = 4x² +4h² +8xh +5x +5h +3

The difference quotient is ...

  (f(x+h) -f(x))/h = ((4x² +4h² +8xh +5x +5h +3) - (4x² +5x +3))/h

  = (4h² +8xh +5h)/h

  (f(x+h) -f(x))/h = 4h +8x +5

When h=0, the value of this is ...

  f'(x) = 4·0 +8x +5

  f'(x) = 8x +5

__

Additional comment

Technically, the difference quotient is undefined at h=0, because h is in the denominator, and we cannot divide by 0. The limit as h→0 will be the value of the simplified rational expression that has h canceled from every term of the difference. This will always be the case for difference quotients for polynomial functions.

<95141404393>

The square root of 184 is approximately 13.5647. It is a non-repeating, non-terminating decimal.

The square root is obtained by finding the number that, when multiplied by itself, equals 184. In this case, 13.5647 multiplied by itself is approximately equal to 184. To explain the answer further, the square root is a mathematical operation that determines the value which, when multiplied by itself, gives the original number.

In the case of 184, the square root is an irrational number, meaning it cannot be expressed as a fraction or a terminating decimal. The approximate value of 13.5647 is derived using numerical methods or a calculator. This value represents the principal square root of 184, and it is positive since the square of a negative number would yield a positive result.

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Answer:

The square root of 184 is expressed as √184 in the radical form and as (184)½ or (184)0.5 in the exponent form. The square root of 184 rounded up to 5 decimal places is 13.56466. It is the positive solution of the equation x2 = 184. We can express the square root of 184 in its lowest radical form as 2 √46.

Square Root of 184: 13.564659966250536

Square Root of 184 in exponential form: (184)½ or (184)0.5

Square Root of 184 in radical form: √184 or 2 √46

The measure of the intercepted arc SU in the circle is 112 degrees.

An inscribed angle is simply an angle with its vertex on the circle and whose sides are chords.

The relationhip between an an inscribed angle and intercepted arc is expressed as:

Inscribed angle = 1/2 × intercepted arc.

From the diagram:

Inscribed angle = 56 degrees

Intercepted arc SU= ?

Plug the given value into the above formula and solve for the intercepted arc.

Inscribed angle = 1/2 × intercepted arc

56 = 1/2 × arc SU

Multiply both sides by 2:

56 × 2 = 1/2 × 2 × arc SU

112 = arc SU

Arc SU = 112°

Therefore, the intercepted arc measure 112 degrees.

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The solution to the polynomial division in quotient and remainder form is: (x + 2) + (-5x + 4)/(x² - 2x + 1)

The polynomials we want to divide are:

x³ - 8x + 6 by x² - 2x + 1 and as such we can write it as:

                x + 2

x² - 2x + 1|x³ - 8x + 6

             -  x³ - 2x² + x

                     2x² - 9x + 6

                  -  2x² - 4x + 2

                            -5x + 4

Thus, the solution expressed in quotient and remainder form is:

(x + 2) + (-5x + 4)/(x² - 2x + 1)

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Answer:

Step-by-step explanation:

To calculate the future value of Francine's 401k account in 10 years, considering an annual contribution of $8,000 and an average annual return of 4%, we can use the formula for the future value of a series of regular payments, also known as an annuity.

The formula for the future value of an annuity is:

FV = P * [(1 + r)^n - 1] / r

Where:

FV is the future value

P is the payment amount

r is the interest rate per period

n is the number of periods

In this case:

P = $8,000 (annual contribution)

r = 4% or 0.04 (annual interest rate)

n = 10 (number of years)

Calculating the future value:

FV = $8,000 * [(1 + 0.04)^10 - 1] / 0.04

FV = $8,000 * (1.04^10 - 1) / 0.04

FV ≈ $8,000 * (1.480244 - 1) / 0.04

FV ≈ $8,000 * 0.480244 / 0.04

FV ≈ $8,000 * 12.0061

FV ≈ $96,048.80

Therefore, Francine will have approximately $96,048.80 in her 401k account in 10 years if the account averages a 4% annual return and she contributes $8,000 each year.

Answer:

A) [tex]\displaystyle y=\left \{ {{x^3-4,\,x\leq 1} \atop {x^2-3,\,x > 1}} \right.[/tex]

Step-by-step explanation:

The first "piece" of the piecewise function, [tex]y=x^3-4[/tex], contains [tex]x=1[/tex] because of the closed dot there.

The second "piece" of the piecewise function, [tex]y=x^2-3[/tex], doesn't contain [tex]x=1[/tex] because of the open dot there.

What occurs between the two pieces is called a jump discontinuity.

Therefore, A is the correct answer.