There are 40 students trying out for a soccer team. After tryouts, 16 of these
students will make the soccer team and 4 of those who make the team will be
captains. What is the probability that Julie will be captain, given that she makes
the team?

The shapes matched to the correct answers are given as follows.

A  = 3  - SAS

B = 1  Not Similar

C = 4 - SSS

D = 2 - AA

SAS stands for "side, angle, side," and it denotes that we have two triangles with two sides and an included angle that are equal.

The triangles are congruent if two sides and the included angle of one triangle are equivalent to the corresponding sides and angle of another triangle.

From A, we can see that ΔFEQ is similar to ΔRSQ because the ration of their sides are similar. That is

EQ/FQ = SQ/RQ

⇒ 36/18 = 48/24

⇒ 2 = 2

Also since they share the same angle EQF and SQR (congruence of opposite angles)

then A  = 3  - SAS

B)  B = 1 - that is both triangles are not similar. If the above principles are applied, we would find that none of the sides have similar ratio.

For example,

48/28 ≠ 36/20

Thus, B =  1  Not Similar

C  = 4 - SSS

this is because all side are similar.

BC: TU = 9:1
BD : SU = 9:1

DC : TS = 9:1

Hence, Side - Side - Side postulate applied.

D. In tis case we are given triangles with two similar angles each so Angle - Angle postulate applies. That is D =  2 - AA

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Answer:

65 units

Step-by-step explanation:

solution Given:

apothem(a)=10

no of side(n)= 10

First, we need to find the length of one side (s).

We can find the length of one side using the following formula:

[tex]\boxed{\bold{s = 2 * a * tan(\frac{\pi}{n})}}[/tex]

substituting value:

[tex]\bold{s = 2 * 10 * tan(\frac{\pi}{10})=6.498}[/tex]   here π is 180°

Now

Perimeter: n*s

substituting value:

Perimeter = 10*6.498= 64.98 in nearest tenth 65 units

Therefore, the Perimeter of a regular decagon is 65 units.

Answer:

65.0 units

Step-by-step explanation:

A regular decagon is a 10-sided polygon with sides of equal length.

To find its perimeter, we first need to find its side length (s).

As we have been given its apothem, we can use the apothem formula to find an expression for side length (s).

[tex]\boxed{\begin{minipage}{5.5cm}\underline{Apothem of a regular polygon}\\\\$a=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\where:\\\phantom{ww}$\bullet$ $s$ is the side length.\\ \phantom{ww}$\bullet$ $n$ is the number of sides.\\\end{minipage}}[/tex]

Given the apothem is 10 units and the number of sides is 10, substitute a = 10 and n = 10 into the formula and solve for s:

[tex]10=\dfrac{s}{2 \tan \left(\dfrac{180^{\circ}}{10}\right)}[/tex]

[tex]10=\dfrac{s}{2 \tan \left(18^{\circ}\right)}[/tex]

[tex]s=20 \tan \left(18^{\circ}\right)[/tex]

The perimeter (P) of a regular polygon is the product of its side length and the number of sides. Therefore, the perimeter of the given regular decagon is:

[tex]P=s \cdot n[/tex]

[tex]P=20 \tan \left(18^{\circ}\right) \cdot 10[/tex]

[tex]P=200 \tan \left(18^{\circ}\right)[/tex]

[tex]P=64.9839392...[/tex]

[tex]P=65.0\; \sf units\;(nearest\;tenth)[/tex]

Therefore, the perimeter of a regular decagon with an apothem of 10 units is 65.0 units, to the nearest tenth.

The statement about the function is: "The function is decreasing for all real values of x where x < -4." D.

The given information tells us that the graph represents a downward-opening parabola.

The vertex of the parabola is located at (-4, 4) indicates that this point is the highest point on the graph.

As we move to the left of the vertex, the function values decrease, indicating a decreasing trend.

Moreover, the graph passes through the point (-6, 0), which lies to the left of the vertex.

This confirms that the function is decreasing for all real values of x less than -4, including x < -6.

On the other hand, the graph also passes through the point (-2, 0), which lies to the right of the vertex.

This does not impact the conclusion that the function is decreasing for x < -4, as the graph's behavior to the right of the vertex is not relevant to this particular statement.

Based on the given information and the properties of the downward-opening parabola, we can conclude that the function is decreasing for all real values of x where x < -4.

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The wheelchair access ramp must be 216.09 inches long.

To find the length of the wheelchair access ramp, we can use trigonometry.

The tangent function relates the angle of elevation to the ratio of the opposite side (height) to the adjacent side (length of the ramp).

Let's denote the length of the ramp as "x".

The height of the ramp is given as 18 inches.

Using the tangent function:

tan(angle of elevation) = height/length of the ramp

tan(4.8 degrees) = 18/x

To solve for x, we can rearrange the equation:

x = 18 / tan(4.8 degrees)

Using a calculator to evaluate the tangent of 4.8 degrees:

x = 18 / 0.08331

x= 216.09

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The equation of the circle with center (-4, 4) and passing through (-2, 4) is [tex](x + 4)^2 + (y - 4)^2 = 4.[/tex]

We have,

To determine the equation of a circle, we need the center coordinates and the radius. The center coordinates are given as (-4, 4), and we have a point on the circle as (-2, 4).

The distance between the center (-4, 4) and the point on the circle (-2, 4) represents the radius of the circle.

Using the distance formula.

[tex]radius = √[(x_2 - x_1)^2 + (y_2 - y_1)^2]\\= \sqrt{(-2 - (-4))^2 + (4 - 4)^2}\\= \sqrt{2^2 + 0^2}\\= \sqrt{4}\\= 2[/tex]

Now that we have the center coordinates (-4, 4) and the radius 2, we can write the equation of the circle in standard form:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

Substituting the values:

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

Therefore,

The equation of the circle with center (-4, 4) and passing through (-2, 4) is [tex](x + 4)^2 + (y - 4)^2 = 4.[/tex]

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The equation of the parabola is (x - 3)² = 20(y - k)

Given data ,

To write the equation of a parabola given its directrix and focus, we can use the standard form for a parabola with a vertical axis of symmetry:

(x - h)² = 4p(y - k)

where (h, k) represents the vertex of the parabola, and the distance between the vertex and the focus is equal to the distance between the vertex and the directrix.

In this case, the directrix is x = -2 and the focus is located at (8, -8). Since the directrix is a vertical line, the parabola has a horizontal axis of symmetry.

And , the vertex lies on the axis of symmetry, which is the line equidistant between the directrix and the focus. In this case, the axis of symmetry is the vertical line x = (8 + (-2)) / 2 = 3.

So, the vertex of the parabola is (3, k), where k is yet to be determined

Next, we need to find the value of p, which represents the distance between the vertex and the focus. Since the focus is at (8, -8), and the vertex is at (3, k), the distance between them is given by

p = 8 - 3 = 5

Now, substituting the values into the standard form equation, we have:

(x - 3)² = 4(5)(y - k)

Simplifying further:

(x - 3)² = 20(y - k)

Hence , the equation of the parabola is (x - 3)^2 = 20(y - k), where k is the y-coordinate of the vertex

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We cannot conclude that there are more than 70,000 defined words in the dictionary.

To test the claim that there are more than 70,000 defined words in the dictionary, we can set up the null and alternative hypotheses as follows:

Null Hypothesis (H0): The mean number of defined words on a page is 48.0 or less.

Alternative Hypothesis (H1): The mean number of defined words on a page is greater than 48.0.

So, sample mean

= (59 + 37 + 56 + 67 + 43 + 49 + 46 + 37 + 41 + 85) / 10

= 510 / 10

= 51.0

and, the sample standard deviation (s)

= √[((59 - 51)² + (37 - 51)² + ... + (85 - 51)²) / (10 - 1)]

≈ 16.23

Next, we calculate the test statistic using the formula:

test statistic = (x - μ) / (s / √n)

In this case, μ = 48.0, s ≈ 16.23, and n = 10.

test statistic = (51.0 - 48.0) / (16.23 / √10) ≈ 1.34

With a significance level of 0.05 and 9 degrees of freedom (n - 1 = 10 - 1 = 9), the critical value is 1.833.

Since the test statistic (1.34) is not greater than the critical value (1.833), we do not have enough evidence to reject the null hypothesis.

Therefore, based on the given data, we cannot conclude that there are more than 70,000 defined words in the dictionary.

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Answer:

Step-by-step explanation:

A = 10 cm - 4cm = 6cm

B =

[tex]\sqrt{4^2+7^2}\\ = \sqrt{16+49}\\ = \sqrt{65}[/tex]

C  =  [tex]\sqrt{10^2+3^2}\\ = \sqrt{109}[/tex]

D = [tex]\sqrt{10^2+6^2}\\ = \sqrt{136}[/tex]

The three consecutive multiples of 3 are 15, 18 and 21

First, let's determine three successive multiples of 3:

The subsequent two would be "x+3" and "x+6" if we call the initial number "x".

Since we are aware that the third number (x+6) is one more than the first two numbers (x + x+3), we can write the following equation:

2/3(x + x+3) = (x+6) + 1

Simplifying this equation, we get:

2/3(2x+3) = x+7

Multiplying both sides by 3, we get:

2(2x+3) = 3(x+7)

Expanding and simplifying, we get:

4x + 6 = 3x + 21

Subtracting 3x and 6 from both sides, we get:

x = 15

Therefore, the three consecutive multiples of 3 are 15, 18 and 21

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The length of the other leg of the right triangle is 13.9 cm.

A right angle triangle is a triangle that has one of its angles as 90 degrees. The side of the right triangle can be named according to it angle position as hypotenuse side, adjacent side and opposite side.

Therefore, let's find the other leg of a right triangle with hypotenuse as 19 cm and one leg as 13 cm using Pythagoras's theorem as follows:

c² = a² + b²

where

Therefore,

b = √19² - 13²

b = √361 - 169

b = √192

b = 13.8564064606

Therefore,

length of the other leg = 13.9 cm

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The transformation of Δ ABC  to ΔADE is a dilation by a scale factor of 1/2 and then 180 degrees rotation about the origin.

We have,

In ΔABC,

The coordinates of each point are:

A = (0, 0)

B = (0, -6)

C = (8, -6)

And,

ΔADE,

A = (0, 0)

D = (0, 3)

E = (4, 3)

Now,

We can see that,

If we use a scale factor 1/2 on each the coordinates of A, B, and C and rotated to 180 degrees about the origin.

We get,

A = (0, 0) = (0, 0)

B = (0, - 6) = (0, -3) = (0, 3)

C = (8, -6) = (4, -3) = (4, 3)

Thus,

The transformation of Δ ABC  to ΔADE is a dilation by a scale factor of 1/2 and then 180 degrees rotation about the origin.

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Zero:

x = 0,

Multiplicity:

1,

Effect:

Intersects x-axis at x = 0

Zero:

x = 1,

Multiplicity:

1,

Effect:

Intersects x-axis at x = 1

Zero:

x = -2/5,

Multiplicity:

3,

Effect:

Intersects x-axis at x = -2/5 =0 with a steep curve

The zeros, multiplicity, and effect of the function f(x) = 3x(x - 1)˚ (5x + 2)³, we need to factor the expression.

First, let's identify the zeros by setting the expression equal to zero and solving for x:

3x(x - 1)˚ (5x + 2)³ = 0

Setting each factor equal to zero:

3x = 0 --> Zero: x = 0

x - 1 = 0 --> Zero: x = 1

(5x + 2)³ = 0 --> Zero: x = -2/5

Now, let's determine the multiplicity and effect of each zero:

Zero:

x = 0

To determine the multiplicity, we look at the exponent of the factor (x) in the expression.

The factor x has a multiplicity of 1 since it appears once.

The effect of this zero is that it is a root of the function and causes the graph to intersect the x-axis at x = 0.

Zero:

x = 1

Similar to the previous case, the factor (x - 1) has a multiplicity of 1 since it appears once.

The effect of this zero is that it is a root of the function and causes the graph to intersect the x-axis at x = 1.

Zero:

x = -2/5

The factor (5x + 2) has a multiplicity of 3 because it appears cubed (exponent of 3).

The effect of this zero is that it is a root of the function and causes the graph to intersect the x-axis at x = -2/5 with a steep curve, as it is raised to the power of 3.

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The slope of the straight line in the graph that expresses proportional relationship indicates that the lines in the graph that have a slope greater  than 1 but less than 2 are;

The slope of the graph of a straight line is the ratio of the rise to the run on the line.

The slope of a graph with a slope of 1 has an increase in the y-value of 1 for each increase in the x-value of 1

Slope = 1 = Δy/Δx

When the slope is greater than 1, we get;

Δy/Δx > 1, therefore Δy is larger than 1 when Δx is 1.

Similarly, the slope of a graph with a slope of 2 has an increase in the y-value of 2 for each increase in the x-value of 1

Slope = 2 = Δy/Δx

When the slope is less than 1, we get;

Δy/Δx < 2, therefore Δy is less than 2 when Δx is

The lines in the graph that have a slope greater than 1 but less than 2 are therefore the graphs with the coordinates;

Line 3; (0, 0), (4, 6); Slope = 6/4 = 3/2, therefore; 1 < slope = 3/2 < 2

Line 4; (0, 0), (5, 6); Slope = 6/5, therefore; 1 < Slope < 2

The line 5 has a slope of 1, and the line 1, has a slope of 3, line 2 has a slope of 2

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The standard deviation of the population function x is 2.

We have,

To find the standard deviation of the population function x, we need the mean M(x) and the mean of the squared values M(x²).

So,

Standard Deviation = √[M(x²) - (M(x))²]

Given that M(x) = 2 and M(x²) = 8, we can substitute these values into the formula:

Standard Deviation = √[8 - (2)²]

Standard Deviation = √[8 - 4]

Standard Deviation = √4

Standard Deviation = 2

Therefore,

The standard deviation of the population function x is 2.

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3√2 and 3 are the values of x and y respectively from the figure.

The given diagram is a right triangle with an acute angle of 45 degrees

We need to determine the values of variables x and y.

Applying the trigonometry identity, we will have:

sin 45 = opposite/hypotenuse
sin45 = 3/x

x = 3/sin45

x = 3/(1/√2)
x = 3√2

Similarly:

tan 45 = opposite/adjacent
tan 45 = 3/y

1 = 3/y

y = 3

Hence the values of x and y from the figure is 3√2 and 3 respectively

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The graph of the piecewise function g(x) is given by the image presented at the end of the answer.

A piece-wise function is a function that has different definitions, depending on the input of the function.

The definitions to the function in this problem are given as follows:

Hence the graph of the piecewise function g(x) is given by the image presented at the end of the answer.

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Using the least-squares regression method, the line of best fit is line d.

According to the graph, we can see that the sum of distance to line d is smallest ( least-square).

Least-squares regression is a statistical technique employed to ascertain the optimal line of best fit within a dataset. This line is determined by minimizing the total sum of squared discrepancies between the observed y-values and the predicted y-values that lie on the line.

In this method, we aim to find the line that best captures the relationship between the variables by minimizing the squared differences. By doing so, we strive to minimize the overall error and maximize the accuracy of the line in representing the data.

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The lateral and the total surface area of the rectangular pyramid are 182 square cm and  278 square cm

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

Length = 6 cm

Width = 8 cm

Height = 6.5 cm

The lateral surface area of the rectangular pyramid is calculated as

LA = 2 *(L + W)H

So, we have

LA = 2 *(6 + 8) * 6.5

Evaluate

LA = 182

The total surface area of the rectangular pyramid is calculated as

TA = 2 * (LW + LH + HW)

So, we have

TA = 2 * (8 * 6 + 8 * 6.5 + 6.5 * 6)

Evalaute

TA = 278

Hence, total surface area of the rectangular pyramid is 278 square cm

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The correct option is B, the total volume is 6 gallons, 2 quarts, and 1 pint.

Here we just need to add all the volumes that are in the question, we have.

So there are:

3 gallons.

8 quarts.

13 pints.

Now let's do the changes of units, we know that:

8 pints = 1 gallon

4 quarts = 1 gallon

1 quart = 2 pints

then:

13 pints = 1 gallon  and 5 pints

8 quarts = 2 gallons

5 pints = 2 quarts and 1 pint

Then the total volume is:

6 gallons, 2 quarts, and 1 pint.

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The required measure of point P above the sea level is 6717.22 feet.

Given that,

A is known to be 6,500 feet above sea level; AB = 230 feet.

And, The angle at A looking up at P is 20°.

The angle at B looking up at P is 35°.

Since, These are the equation that contains trigonometric operators such as sin, cos.. etc.. In algebraic operations.

Here,

Let the horizontal distance between A and Q be x and the vertical distance between P and Q be h.

Now,

tan20 = h / x

034x = h

Now,

tan35 = [230 + h]/x

0.7x = 230 + 0.34x

x = 636.89

Now,

h = 0.34 x 636.89

h = 217.2

Now,

The measure of peak P above sea level = 217.2 + 6500 = 6717.22

Thus, the required measure of point P above the sea level is 6717.22 feet.

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Answer:

D

Step-by-step explanation:

sin  (  3pi / 4 )

 =  sin   (   pi   -   pi  /  4   )

 =  sin   (   pi / 4  )

 = 1/root(2)

  =  root(2) / 2

Each number should be dragged to a box to complete the table as follows;

Kilometers     Meters

1                        1,000

2                       2,000

3                       3,000

5                       5,000

8                       8,000

In Science and Mathematics, a conversion factor can be defined as a number that is used to convert a number in one set of units to another, either by dividing or multiplying.

Generally speaking, there are one (1) kilometer in one thousand (1,000) meters. This ultimately implies that, a proportion or ratio for the conversion of kilometer to meters would be written as follows;

Conversion:

1 kilometer = 1,000 meters

2 kilometer = 2,000 meters

3 kilometer = 3,000 meters

4 kilometer = 4,000 meters

5 kilometer = 5,000 meters

6 kilometer = 6,000 meters

7 kilometer = 7,000 meters

8 kilometer = 8,000 meters

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Using  the associative law of multiplication we get this two solution of the given expression: (12x)y = (12 * x) * y , (12x)y = 12 * (x * y)

The associative law of multiplication states that when multiplying three or more numbers, the product is the same regardless of the order in which the numbers are grouped. In other words, it doesn't matter which numbers we multiply first, the result will be the same.

Using the associative law of multiplication, we can group the factors in any way we like. Therefore, we can write an equivalent expression to (12x)y by changing the grouping of factors.

One way to group the factors is to group the two numerical coefficients (12 and y) together.

Another way to group the factors is to group the variable x and the coefficient y together.

Both of these expressions are equivalent to (12x)y and they all follow the associative law of multiplication.

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To find the number of buttons in each of Jack's five bags, we can use the information given to us and solve for the unknowns. Let's start by finding the total number of buttons:

Total number of buttons = Mean number of buttons × Number of bags

Total number of buttons = 27 × 5

Total number of buttons = 135

Since we know the fewest and greatest number of buttons in a bag, we can find the range:

Range = Greatest number of buttons - Fewest number of buttons

Range = 29 - 21

Range = 8

Now, we can set up two equations to find the two modes:

Mode 1 + Mode 2 + 3 × 27 = 135 (sum of all bags)

Mode 2 - Mode 1 = 8 (difference between the two modes)

Solving these equations simultaneously, we get:

Mode 1 = 23

Mode 2 = 31

Therefore, the number of buttons in each of Jack's five bags is:

Bag 1: 21 buttons

Bag 2: 23 buttons

Bag 3: 25 buttons

Bag 4: 27 buttons

Bag 5: 29 buttons

The measurement that is closest to the volume of the sphere is given as follows:

1,767.1 in³.

The volume of an sphere of radius r is given by the multiplication of 4π by the radius cubed and divided by 3, hence the equation is presented as follows:

[tex]V = \frac{4\pi r^3}{3}[/tex]

From the image given at the end of the answer, we have that the diameter is of 15 units, hence the radius of the sphere, which is half the diameter, is given as follows:

r = 0.5 x 15

r = 7.5 units.

Then the volume of the sphere is given as follows:

V = 4/3 x π x 7.5³

V = 1,767.1 in³.

The sphere is given by the image presented at the end of the answer.

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Int he above expression, A = 3 (distance from the vertex to the center)

B = 4

C = 3 (distance from focus to center)

D = 10

Since the center of the hyperbola is (3,10), we have C=3 and D=10.

The distance from the center to the vertex is A, so we have A= 6-3

A = 3.

The distance from the center to the focus is given by c, so we have c=8-3=5.

We can use the relationship a² + b² = c² to solve for B:

a = 6 - 3 = 3 (distance from vertex to center)

c = 5 (distance from focus to center)

b = ?

b² = c² - a²

b² = 5² - 3²

b² = 16

b = 4

Therefore, the equation of the hyperbola is

((x-3)²/3²) - ((y-10)²/4²) = 1

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Full Quesitn:

The equation of the hyperbola that has a center at (3, 10), a focus at (8, 10), and a vertex at (6, 10), is

((x-C)²/A²) - ((y-D)²)/B²) = 1

Where A = ?

B  = ?

C = ?

D = ?

The total mass of the packages is 37.67 kg

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

Packages    Mass (kg)

1                         3.94

2                      14.81

3                          11.27

4                           7.65

The total mass of the packages is the sum of each mass

using the above as a guide, we have the following:

Total mass = 3.94 + 14.81 + 11.27 + 7.65

Evaluate the sum

So, we have

Total mass = 37.67

Hence, the total mass of the packages is 37.67 kg

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The values of Composite function are:

f(g(-1)) = 20

h[g(4)] = 40

g[f(5)] = -37

f[h(-4)] = -44

g[g(7)] = 35

h[f(1/2)] = -8

We have, f(x) = -2x, g(x) = 3x-7 and h(x)= 2x² -10

f(g(-1)):

First, substitute -1 into g(x): g(-1) = 3(-1) - 7 = -3 - 7 = -10.

Next, substitute the result into f(x): f(-10) = -2(-10) = 20.

Therefore, f(g(-1)) = 20.

h[g(4)]:

First, substitute 4 into g(x): g(4) = 3(4) - 7 = 12 - 7 = 5.

Next, substitute the result into h(x): h(5) = 2(5^2) - 10 = 2(25) - 10 = 50 - 10 = 40.

Therefore, h[g(4)] = 40.

g[f(5)]:

First, substitute 5 into f(x): f(5) = -2(5) = -10.

Next, substitute the result into g(x): g(-10) = 3(-10) - 7 = -30 - 7 = -37.

Therefore, g[f(5)] = -37.

f[h(-4)]:

First, substitute -4 into h(x): h(-4) = 2(-4²) - 10 = 2(16) - 10 = 32 - 10 = 22.

Next, substitute the result into f(x): f(22) = -2(22) = -44.

Therefore, f[h(-4)] = -44.

g[g(7)]:

First, substitute 7 into g(x): g(7) = 3(7) - 7 = 21 - 7 = 14.

Next, substitute the result into g(x) again: g(14) = 3(14) - 7 = 42 - 7 = 35.

Therefore, g[g(7)] = 35.

h[f(1/2)]:

First, substitute 1/2 into f(x): f(1/2) = -2(1/2) = -1.

Next, substitute the result into h(x): h(-1) = 2(-1²) - 10 = 2(1) - 10 = 2 - 10 = -8.

Therefore, h[f(1/2)] = -8.

Learn more about Composite function here:

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