The cactus grill has decided to run a special on Wednesday night where every table gets 10% off their bill find the total for this table

1) 0.080 grams of 226Ra would remain after 17,000 years.

2) 44.000 grams of 226Ra would remain after 1,700 years.

To calculate the remaining amount of a radioactive isotope after a certain time, we can use the formula:

[tex]N(t) = N₀ * (1/2)^{(t / T_{ \frac{1}{2}} )}[/tex]

Where:

N(t) is the remaining amount of the isotope after time t.

N₀ is the initial amount of the isotope

[tex]T_{ \frac{1}{2} }[/tex] is the half-life of the isotope

Let's calculate the remaining amount of 226Ra for the given time periods:

(1) After 1,700 years:

[tex]N(t) = 80g * (1/2)^(1700 / 1599) \\ N(t) = 80g *(1/2)^(1.063165727329581) \\ N(t) ≈ 80g * 0.550 \\ N(t) ≈ 44.000g[/tex]

(rounded to three decimal places)

Therefore, approximately 44.000 grams of 226Ra would remain after 1,700 years.

(2) After 17,000 years:

[tex]N(t) = 80g * (1/2)^(17000 / 1599) \\ N(t) = 80g * (1/2)^(10.638857911194497) \\ N(t) ≈ 80g * 0.001 \\ N(t) ≈ 0.080g [/tex]

(rounded to three decimal places)

Therefore, approximately 0.080 grams of 226Ra would remain after 17,000 years.

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If the null space of a 9x4 matrix A is 3-dimensional, the dimension of the row space of A is 1.

If the null space of a 9x4 matrix A is 3-dimensional, the dimension of the row space of A can be found using the Rank-Nullity Theorem.

The Rank-Nullity Theorem states that for a matrix A with dimensions m x n, the sum of the dimension of the null space (nullity) and the dimension of the row space (rank) is equal to n, which is the number of columns in the matrix. Mathematically, this can be represented as:

rank(A) + nullity(A) = n

In your case, the null space is 3-dimensional, and the matrix A has 4 columns, so we can write the equation as:

rank(A) + 3 = 4

To find the dimension of the row space (rank), simply solve for rank(A):

rank(A) = 4 - 3
rank(A) = 1

So, if the null space of a 9x4 matrix A is 3-dimensional, the dimension of the row space of A is 1.

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To solve this problem, we can use the relationship between the sides of an isosceles right triangle. Let's denote the length of the hypotenuse as h.

The area of the triangle is given by A = (1/2) * s^2, where s is the length of the legs.

We are given that the area A is increasing at a rate of 12 square centimeters per second. So, we have dA/dt = 12.

Differentiating the area equation with respect to time, we get:

dA/dt = (1/2) * 2s * ds/dt

Since the triangle is isosceles, the two legs have the same length, so we can substitute s for both legs:

12 = s * ds/dt

Now we need to find the rate at which the length of the hypotenuse h is changing with respect to time, dh/dt.

Using the Pythagorean theorem, we know that h = sqrt(2) * s.

Differentiating the equation with respect to time, we get:

dh/dt = (d/dt)(sqrt(2) * s)

Using the chain rule, we have:

dh/dt = sqrt(2) * ds/dt

Substituting the value of ds/dt from the earlier equation, we have:

dh/dt = sqrt(2) * (12/s)

At the instant when s = sqrt(32), we can substitute this value into the equation:

dh/dt = sqrt(2) * (12/sqrt(32))

Simplifying, we have:

dh/dt = sqrt(2) * (12/4)

dh/dt = sqrt(2) * 3

Therefore, at that instant, the length of the hypotenuse of the triangle is increasing at a rate of 3 * sqrt(2) centimeters per second.

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The premises is R • ∼E • ∼D • G

This is the desired conclusion.

The premises, we can conclude that:

R • ∼E • ∼D

The following steps of deductive reasoning:

From premise 3 and the contrapositive of premise 1 can deduce that:

∼(E • D) ⊃ ∼R

Using De Morgan's Law can rewrite this as:

(∼E ∨ ∼D) ⊃ ∼R

Since R ⊃ (E • D) by premise 1 can substitute this into the above equation to get:

(∼E ∨ ∼D) ⊃ ∼(R ⊃ (E • D))

Using the rule of implication can simplify this to:

(∼E ∨ ∼D) ⊃ (R • ∼(E • D))

From premise 2 know that R • ∼G.

Using De Morgan's Law can rewrite this as:

∼(R ∧ G)

Combining this with the above equation get:

(∼E ∨ ∼D) ⊃ ∼(R ∧ G ∧ E ∧ D)

Simplifying this using De Morgan's Law and distributivity get:

(∼E ∨ ∼D) ⊃ (∼R ∨ ∼G)

Finally, using premise 3 and modus ponens can deduce that:

∼E ∨ ∼D ∨ G

Since we know that R • ∼G from premise 2 can substitute this into the above equation to get:

∼E ∨ ∼D ∨ ∼(R • ∼G)

Using De Morgan's Law can simplify this to:

∼E ∨ ∼D ∨ (R ∧ G)

Multiplying both sides by R and ∼E get:

R∼E∼D ∨ R∼EG

Using distributivity and commutativity can simplify this to:

R(∼E∼D ∨ ∼EG)

Finally, using De Morgan's Law can rewrite this as:

R(∼E ∨ G) (∼D ∨ G)

This is equivalent to:

R • ∼E • ∼D • G

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A sample size of approximately 136 is needed to estimate the true mean score within 5 points with 95% confidence.

To determine the sample size needed to estimate the true mean score within 5 points with 95% confidence, we can use the formula for sample size calculation:

n = (Z * σ / E)²

In this case, the standard deviation (σ) of the statistics test is given as 29.7, and the desired margin of error (E) is 5.

Plugging these values into the formula:

[tex]n = (1.96 * 29.7 / 5)^2[/tex]

Calculating this expression:

n ≈[tex](58.212 / 5)^2[/tex]

n ≈ [tex]11.6424^2[/tex]

n ≈ 135.6336

Therefore, a sample size of approximately 136 is needed to estimate the true mean score within 5 points with 95% confidence.

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Possible subsets of A using two elements are:

{a, b}, {a, c}, {a, d}, {a, e},

{b, c}, {b, d}, {b, e},

{c, d}, {c, e},

{d, e}

Possible arrangements of these two elements are:

ab, ac, ad, ae,

bc, bd, be,

cd, ce,

de

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The angle the ladder makes with the ground is approximately 58.1 degrees.

We can utilize geometry to find the point that the stepping stool makes with the ground. We should call the point we need to find "theta" (θ).

In the first place, we can draw a right triangle with the stepping stool as the hypotenuse, the separation from the wall as the contiguous side, and the level the stepping stool comes to as the contrary side. Utilizing the Pythagorean hypothesis, we can track down the level of the stepping stool:

[tex]a^2 + b^2 = c^2[/tex]

where an is the separation from the wall (6 m), b is the level the stepping stool ranges, and c is the length of the stepping stool (11 m). Improving the condition and settling for b, we get:

b = [tex]\sqrt (c^2 - a^2)[/tex] = [tex]\sqrt(11^2 - 6^2)[/tex] = 9.3 m

Presently, we can utilize the digression capability to track down the point theta:

tan(theta) = inverse/contiguous = b/a = 9.3/6

Taking the converse digression (arctan) of the two sides, we get:

theta = arctan(9.3/6) = 58.1 degrees (adjusted to one decimal spot)

Subsequently, the point that the stepping stool makes with the ground is around 58.1 degrees.

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Answer:

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

Area is the product of two dimensions, so the ratio of areas of similar figures is equal to the square of the scale factor k.

Hence the scale factor is:

Therefore the ratio of corresponding parts is:

The length JL in the similar triangle is 17.5 units.

Similar triangles are the triangles that have corresponding sides in proportion to each other and corresponding angles equal to each other.

Therefore, let's use the proportional relationships to find the length JL of the triangle as follows:

Hence, using the proportion,

GH / KJ = GI / JL

Therefore,

12 / 30 = 7 / JL

cross multiply

12 JL =  210

divide both sides by 12

JL = 210 / 12

JL = 17.5 units

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Simplified results are after substitute the f(x, y) = 2x - y + 5 : (a) 3 (b) 3 (c) -15 (d) 11-y (e) 2x + 1 (f) 14

a) To evaluate f(0,2), we simply substitute x = 0 and y = 2 into the expression for f(x,y):
f(0,2) = 2(0) - 2 + 5 = 3
So, f(0,2) simplifies to 3.
b) To evaluate f(-1,0), we substitute x = -1 and y = 0:
f(-1,0) = 2(-1) - 0 + 5 = 3
So, f(-1,0) simplifies to 3 as well.
c) To evaluate f(5,30), we substitute x = 5 and y = 30:
f(5,30) = 2(5) - 30 + 5 = -15
So, f(5,30) simplifies to -15.
d) To evaluate f(3,y), we substitute x = 3 and leave y as y:
f(3,y) = 2(3) - y + 5 = 11 - y
So, f(3,y) simplifies to 11 - y.
e) To evaluate f(x,4), we substitute y = 4 and leave x as x:
f(x,4) = 2x - 4 + 5 = 2x + 1
So, f(x,4) simplifies to 2x + 1.
f) To evaluate f(5,1), we substitute x = 5 and y = 1:
f(5,1) = 2(5) - 1 + 5 = 14
So, f(5,1) simplifies to 14.

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To find f, we need to integrate the given second derivative of f twice. We start by integrating the second derivative with respect to x to obtain the first derivative:

f'(x) = ∫f ″(x) dx = 24x^(4)/4 - 18x^(3)/3 + 10x^2/2 + c_1

      = 6x^4 - 6x^3 + 5x^2 + c_1

where c_1 is a constant of integration.

Next, we integrate f'(x) with respect to x to obtain f(x):

f(x) = ∫f'(x) dx = ∫[6x^4 - 6x^3 + 5x^2 + c_1] dx

       = 6x^(5)/5 - 6x^(4)/4 + 5x^(3)/3 + c_1x + c_2

where c_2 is a constant of integration.

Therefore, the solution for f is:

f(x) = 6x^(5)/5 - 6x^(4)/4 + 5x^(3)/3 + c_1x + c_2

where c_1 and c_2 are constants of integration.

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Answer:

6.4 inches

Step-by-step explanation:

The diagonal of a square is equal to the diameter of the circle that can be inscribed in the square. So, if we can find the diameter of the circle, we can then find the length of the square's diagonal, which is also the largest possible length of a side of the square.

The diameter of the circle is 9 inches, so the radius is 4.5 inches. The diagonal of the square is the diameter of the circle, which is 9 inches.

Let's use the Pythagorean theorem to find the length of a side of the square:

a^2 + b^2 = c^2

where a and b are the sides of the square and c is the diagonal.

We know c = 9, so:

a^2 + b^2 = 9^2 = 81

Since we want the largest possible length of a side of the square, we want to maximize the value of a. In a square, a and b are equal, so we can simplify the equation to:

2a^2 = 81

a^2 = 40.5

a ≈ 6.4 (rounded to the nearest tenth of an inch)

Therefore, the largest possible length of a side of the square is approximately 6.4 inches.

True. Triangulation can be used to find the location of an object by measuring the angles.

Triangulation is a method used to determine the location of an object by measuring the angles between the object and two or more reference points whose locations are known.

This method is widely used in surveying, navigation, and various other fields.

By measuring the angles, the relative distances between the object and the reference points can be determined, and then the location of the object can be calculated using trigonometry.

Triangulation is commonly used in GPS systems, where the location of a GPS receiver can be determined by measuring the angles between the receiver and several GPS satellites whose locations are known.

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The arc of the satellite refers to the path that a satellite follows as it orbits around a celestial body such as the Earth, and is determined by the gravitational forces between the two objects.

We are given the orbital radius and angular separation between two adjacent communication satellites, and we need to find the arc length that separates them.

Here's a step-by-step explanation:

1. Given the orbital radius (r) is 8.46 * 10^7 m.
2. Given the angular separation (θ) is 4.00 degrees.
3. To find the arc length (s), we can use the formula s = r * θ, where θ should be in radians.
4. Convert the angular separation from degrees to radians: θ (radians) = θ (degrees) * (π / 180) = 4 * (π / 180) = 4π / 180 radians.
5. Calculate the arc length: s = r * θ = (8.46 * 10^7) * (4π / 180) ≈ 5.92 * 10^6 m.

So, the arc length that separates the satellites is approximately 5.92 * 10^6 m (option b).

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The parameterization of the ellipse in a counterclockwise manner is x = 3cos(t) y = 2sin(t) where t varies from 0 to 2π.

One common way to parameterize an ellipse is to use trigonometric functions such as sine and cosine. We can write the equation of the ellipse as:

4x² + 9y² = 36

We can then use the following parameterization:

x = 3cos(t) y = 2sin(t)

where t is the parameter that varies between 0 and 2π, traversing the ellipse once in a counterclockwise manner.

To see why this parameterization works, let's substitute x and y into the equation of the ellipse:

4(3cos(t))² + 9(2sin(t))² = 36

Simplifying this equation gives:

36cos²(t) + 36sin²(t) = 36

Which is true for any value of t. This shows that our parameterization does indeed describe the ellipse 4x² + 9y² = 36.

Furthermore, we can see that when t=0, we get x=3 and y=0, which is the starting point (3,0). As t varies from 0 to 2π, x and y will trace out the ellipse exactly once in a counterclockwise manner.

Therefore, the parameterization of the ellipse 4x² + 9y² = 36 that begins at the point (3,0) and traverses once in a counterclockwise manner is:

x = 3cos(t) y = 2sin(t)

where t varies from 0 to 2π.

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The component form of vector a⃗, rounded to the nearest hundredth, is:

a⃗ = 〈-12.99, 19.97〉

To find the component form of vector a⃗, which is expressed in magnitude and direction form as a⃗ =〈26√,140°〉, we can use the formulas for converting polar coordinates to rectangular coordinates:

x = r * cos(θ)
y = r * sin(θ)

In this case, r (magnitude) is equal to 26√ and θ (direction) is equal to 140°. Let's calculate the x and y components:

x = 26√ * cos(140°)
y = 26√ * sin(140°)

Note that we need to convert the angle from degrees to radians before performing the calculations:

140° * (π / 180) ≈ 2.4435 radians

Now, let's plug in the values:

x ≈ 26√ * cos(2.4435) ≈ -12.99
y ≈ 26√ * sin(2.4435) ≈ 19.97

Therefore, the component form of vector a⃗ is:

a⃗ = 〈-12.99, 19.97〉

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The variation of parameter formula is used to determine the general solution of a second-order linear differential equation. In this case, we have t^2y''+3ty'+yln(t)=0. To use the variation of parameter formula, we first need to find the complementary solution. Then we can find two particular solutions and use them to form the general solution. The complementary solution is y_c=c1t^(-1/3)+c2t. To find the particular solutions, we assume y1=u1(t)t^(-1/3) and y2=u2(t)t, where u1(t) and u2(t) are functions of t. Plugging these into the differential equation and solving for u1(t) and u2(t), we get the particular solutions. The general solution is then y=y_c+y1+y2.

The given differential equation is t^2y''+3ty'+yln(t)=0. We first find the complementary solution by assuming y=e^(rt) and solving the characteristic equation r^2+3r+ln(t)=0. The roots are complex, so the complementary solution is y_c=c1t^(-1/3)+c2t.
Next, we assume y1=u1(t)t^(-1/3) and y2=u2(t)t as the particular solutions. Then, we can find the derivatives y1'=-u1'(t)t^(-1/3)+(-1/3)u1(t)t^(-4/3) and y2'=u2'(t)t+(1/t)u2(t), and y1''=u1''(t)t^(-1/3)+(2/9)u1(t)t^(-7/3)+(2/3)u1'(t)t^(-4/3) and y2''=u2''(t)t+(2/t)u2'(t)-(1/t^2)u2(t). Plugging these into the differential equation, we get the system of equations:
u1''(t)t^(-1/3)+(2/9)u1(t)t^(-7/3)+(2/3)u1'(t)t^(-4/3)+u2''(t)t+(2/t)u2'(t)-(1/t^2)u2(t)=ln(t)
(-1/3)u1'(t)t^(-1/3)+(1/t)u2(t)=0
Solving for u1(t) and u2(t), we get:
u1(t)=(1/18)t^2(ln(t)-6C1t^(4/3)+18C2t^(2/3))
u2(t)=C3t+((1/3)t^2+C4)ln(t)
Therefore, the general solution is:
y=c1t^(-1/3)+c2t+(1/18)t^2(ln(t)-6C1t^(4/3)+18C2t^(2/3))+C3t+((1/3)t^2+C4)ln(t)

Using the variation of parameter formula, we found the general solution of the given differential equation to be y=c1t^(-1/3)+c2t+(1/18)t^2(ln(t)-6C1t^(4/3)+18C2t^(2/3))+C3t+((1/3)t^2+C4)ln(t). This formula can be used to solve similar second-order linear differential equations.

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The probability of selecting three marbles of the same color from the jar is 54/990, which can be simplified to 3/55

To calculate the probability of selecting three marbles of the same color, we need to consider each color separately.

The probability of selecting three red marbles can be calculated as the product of selecting the first red marble (4/11), the second red marble (3/10), and the third red marble (2/9) without replacement. This gives us (4/11) * (3/10) * (2/9) = 24/990.

Similarly, the probability of selecting three blue marbles is (3/11) * (2/10) * (1/9) = 6/990.

Lastly, the probability of selecting three green marbles is (4/11) * (3/10) * (2/9) = 24/990.

Adding up the probabilities for each color, we have (24/990) + (6/990) + (24/990) = 54/990.

Therefore, the probability of selecting three marbles of the same color from the jar is 54/990, which can be simplified to 3/55.

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Answer:

17.4

Step-by-step explanation:

The Hundredths place is above four so it has to be the next number up

Answer:17.40

Step-by-step explanation:

17.37 rounded to the nearest tenth is, 17.40, because when rounding, you see if the number is 5 or up ( that means you round it up.)

Evaluating this integral yields the volume of the region E.

To find the volume of the region E that lies between the paraboloid x² + y² - z=24 and the cone z = 2 = 2.1x + y, we can use cylindrical coordinates.

The first step is to rewrite the equations in cylindrical coordinates. We can use the following conversions:

x = r cos θ

y = r sin θ

z = z

Substituting these into the equations of the paraboloid and cone, we get:

r² - z = 24

z = 2.1r cos θ + r sin θ

We can now set up the integral to find the volume of the region E. We need to integrate over the range of r, θ, and z that covers the region E. Since the cone and paraboloid intersect at z = 0, we can integrate over the range 0 ≤ z ≤ 24. For a given value of z, the cone intersects the paraboloid when:

r² - z = 2.1r cos θ + r sin θ

Solving for r, we get:

r = (z + 2.1 cos θ + sin θ)/2

Since the cone intersects the paraboloid at r = 0 when z = 0, we can integrate over the range:

0 ≤ θ ≤ 2π

0 ≤ z ≤ 24

0 ≤ r ≤ (z + 2.1 cos θ + sin θ)/2

The volume of the region E is then given by the triple integral:

∭E dV = ∫₀²⁴ ∫₀²π ∫₀^(z+2.1cosθ+sinθ)/2 r dr dθ dz

Evaluating this integral yields the volume of the region E.

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Answer: 3

Step-by-step explanation:

Answer:

2x+9=15

Step-by-step explanation:

Find common denominator

Combine fractions with common denominator

Multiply the numbers

Multiply all terms by the same value to eliminate fraction denominators

Cancel multiplied terms that are in the denominatorMultiply the numbers

The correct choice is: Part of the man’s work goes into the kinetic energy of the crate, while the other part of his work is done against friction.

When the man accelerates the crate across the rough floor, the efficiency of 50% indicates that only half of the work done by the man is effectively transferred to the crate. The remaining half of the work is lost or dissipated, likely due to friction between the crate and the floor.

Therefore, part of the man's work goes into increasing the kinetic energy of the crate, allowing it to gain speed and move. The other part of the man's work is used to overcome the frictional forces acting against the crate's motion.

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Therefore, we can estimate that the landfill contains approximately 1725 tons of solid waste at the end of the first 3 months of 2010.

Using the given information, we know that at t=0 (the beginning of 2010), W=1400 tons. We also know that the differential equation that models the increase in solid waste is dW/dt = 1(W-300).
To approximate the amount of solid waste at the end of the first 3 months of 2010, we need to find the value of W at t=0.25 (since t is measured in years from the start of 2010).
Using the line tangent to the graph of W at t=0, we can estimate the value of W at t=0.25. The slope of the tangent line is equal to dW/dt at t=0, which is 1(1400-300) = 1100 tons/year.
So the equation of the tangent line at t=0 is W = 1400 + 1100(t-0) = 1400 + 1100t. Plugging in t=0.25, we get W=1725 tons.
Using the given differential equation and tangent line, we estimate that the landfill contains approximately 1725 tons of solid waste at the end of the first 3 months of 2010, based on an initial amount of 1400 tons at the beginning of the year.

Therefore, we can estimate that the landfill contains approximately 1725 tons of solid waste at the end of the first 3 months of 2010.

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A door is painted pink and blue. The area painted pink is 4 times that of the area painted blue. To complete the table for July and August, we need to find the changes in the water level for those months.

Given that the total change in the water level from April to August is -4.7 inches, we can use this information to find the changes in the water level for July and August.

By examining the table, we can observe that the changes in the water level for each month are cumulative. To find the changes for July and August, we need to subtract the changes from the previous months from the total change of -4.7 inches.

Let's denote the change in the water level for July as "x" inches. Then, the change for August would be (-4.7 - x) inches since the total change should add up to -4.7 inches.

We don't have specific information to determine the exact values of x and (-4.7 - x), but completing the table would involve finding reasonable values that fit the given total change.

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The vertical direction moved by the graph is 1 unit up

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

y = 7cos(2π/7(x + 9)) + 1

A sinusoidal function is represented as

f(x) = Acos(B(x + C)) + D or

f(x) = Asin(B(x + C)) + D

Where

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

Vertical shift = D = 1

Hence, the vertical direction of the graph is 1 unit up

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when the teacher rewarded 6 students, the total number of tickets passed out would be 6 students multiplied by 2 tickets per student, which equals 12 tickets.

To find out how many tickets were passed out when the teacher rewarded 6 students, we need to examine the given table. We notice that the number of tickets passed out is constant for each day, meaning that the same number of tickets is given to each student.

From the table, we can see that when 3 students were rewarded, 6 tickets were passed out. Similarly, when 2 students were rewarded, 4 tickets were passed out. When 5 students were rewarded, 10 tickets were passed out.

Since the number of tickets passed out is constant for each day, we can determine the number of tickets per student by finding the average number of tickets per student across different days.

Calculating the average, we get (6 + 4 + 10) / (3 + 2 + 5) = 20 / 10 = 2 tickets per student.

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The parametric equations of the line are:

x = 2

y = 1 - t

z = t

And the symmetric equations of the line are:

x - 2 = 0

y - 1 = -1

z = 1

For the line through the point (2, 1, 0) and perpendicular to both i + j and j + k, we can determine the direction vector of the line.

First, let's find the direction vector by taking the cross product of the vectors i + j and j + k:

(i + j) × (j + k) = i × j + i × k + j × j + j × k

= k - i + 0 + i - j + 0

= -j + k

Therefore, the direction vector of the line is -j + k.

Now, we can write the parametric equations of the line using the given point (2, 1, 0) and the direction vector:

x = 2 + 0t

y = 1 - t

z = 0 + t

The parameter t represents a scalar that can vary, and it determines the points on the line.

To write the symmetric equation, we can use the direction vector -j + k as the normal vector. The symmetric equation is given by:

(x - 2)/0 = (y - 1)/(-1) = (z - 0)/1

Simplifying this equation, we get:

x - 2 = 0

y - 1 = -1

z - 0 = 1

Which can be written as:

x - 2 = 0

y - 1 = -1

z = 1

In summary, the parametric equations of the line are:

x = 2

y = 1 - t

z = t

And the symmetric equations of the line are:

x - 2 = 0

y - 1 = -1

z = 1

These equations describe the line that passes through the point (2, 1, 0) and is perpendicular to both i + j and j + k.

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Answer: Here you go champ.

Step-by-step explanation:

a.) FROM LEFT TO RIGHT

5, -3, -4, 0

b.) Curve A

ii.) about -1.75

iii.) 3.236, -1.236

Answer:

a)
Missing values

x = -2        y = 5

x = 0         y = - 3

x = 1          y = - 1

x = 3         y = 0

b)

i)  A

ii) - 1.75

iii)  x = 1 + √5 and x = 1 - √5

In decimal that would be
x = 3.23606 and x = −1.23606

I am not sure which form they want it in

Step-by-step explanation:

Given function is
y = x² - 2x - 3

a) To find the missing y values, plug in the corresponding value of x and solve for y

x = -2  ==> y = (-2)² -2(-2) - 3 y = = 4 + 4 - 3 or y = 5
x = 0, y = 0² -2(0) - 3 = - 3
x = 1, y = 1² -2(1) - 3 = 1 - 2 -3 = - 4
x = 3, y = 3² - 2(3) - 3 = 0

b)

i) Graph A(blue) matches the function expression; when x = 1, y = -4 in this curve

ii) Estimate the value of y when x = 2.5

Plug in x = 2.5 into the function

y = (2,5)² -2(2.5) - 3 = 6.25 - 5 - 3 = - 1.75

c) Find the value of x at y =1

When y = 1, we get
x² - 2x - 3 = 1

Moving 1 to the left side gives

x² - 2x - 4 = 0

This is a quadratic equation which can be solved using the quadratic equation. There are calculators for this to ease your pain

But if doing manually

A quadratic equation of the form
ax² + bx + c = 0 will have the solutions

[tex]x_{1,\:2}=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

In the given expression,
a = 1, b = -2 and c = -4

Let's calculate the individual terms in the quadratic formula and combine everything to solve

b² - 4c = (-2)² -4(1)(-4)
= 4 + 16

= 20

[tex]\sqrt{b^2- 4ac} = \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}[/tex]

Therefore the two values of x are

[tex]x_1=\dfrac{-\left(-2\right)+2\sqrt{5}}{2\cdot \:1}\\\\= \dfrac{2 + 2\sqrt{5}}{2}\\= 1 + \sqrt{5}\\\\[/tex]

[tex]x_2=\dfrac{-\left(-2\right) - 2\sqrt{5}}{2\cdot \:1}\\\\= \dfrac{2 - 2\sqrt{5}}{2}\\\\= 1 - \sqrt{5}\\\\[/tex]

So the values of x when y = 1 are
[tex]x=1+\sqrt{5},\:x=1-\sqrt{5}[/tex]

(a) The value of the function f.g(x) = f(x²+9) = (x²+9)²
(b) The value of the function g.f(x) = g(x²) = x⁴+9
(c) The value of the function f.f(x) = f(x²) = (x²)²
(d) The value of the function g.g(x) = g(x²+9) = (x²+9)²+9

Domains of each:
(a) All real numbers
(b) All real numbers
(c) All real numbers
(d) All real numbers

For composite functions, you insert the second function into the first function.

(a) f.g(x) = f(g(x)) = f(x²+9) = (x²+9)²
(b) g.f(x) = g(f(x)) = g(x²) = x⁴+9
(c) f.f(x) = f(f(x)) = f(x²) = (x²)²
(d) g.g(x) = g(g(x)) = g(x²+9) = (x²+9)²+9

The domain of a function is the set of input values for which the function is defined. Since all these composite functions are polynomial functions, they are defined for all real numbers.

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