Im really struggling and need help with this

1. One of the tables below contains (x,y) values that were generated by a linear function. Determine which table, and then write the equation of the linear function represented by the

Table #1 x 2 5 8 11 14 17 20 y 1 3 7 13 21 31 43

Table #2 x 1 2 3 4 5 6 7 y 10 13 18 21 26 29 34

Table #3 x 2 4 6 8 10 12 14 y 1 6 11 16 21 26 31

2. Give an example of each of the following:

A function whose domain is [2, infinity)
An arithmetic sequence
A system of equations with no solutions

3. For f(x) = 1/x^2-3, find:

f(3)
f(2+h)
4. For f(x) = 1/x-5 and g(x) = x^2+2, find:
1. (fog)(x)
2. (gof)(6)
5. For the sequence given by a (subscript n) = 4n+5 an=4n+5, answer the following:

Find the first five terms
Find the sum of the first 25 terms.
Is this an arithmetic sequence? If so, how can you tell? If not, why not?
6. Graph the area bound by y<1/2x+6, x+3y>=12, x>=0, and x<=12
7. For the function defined by:
f(x)= {x^2, x<=1}
{2x+1, x>1}
a. evaluate f(0)
b. graph f(x)
8. Solve the following system of equations algebraically. Verify your solution either graphically or by using matrices.
3x-y=0
5x+2y=22
9. Solve the following system of equations algebraically. Verify your solution either graphically or by using matrices
8x-2y=5
-12x+3y=7
10.Given matrices A, B, and C below, perform the indicated operations if possible. If the operation is not possible, explain why.
A (3x3) [1st 2, -1, 0] [2nd 0, 5, 0.3] [3rd 1, 4, 10]
B (3x3) [1st 5, 0, 2] [2nd 1, -3, 9] [3rd2, 0, 4]
C (1x1) [1, 3, 5]
1.3A+B
2. B+C
3. CA
11. Given the table below, evaluate the following:

x

-3

-2

-1

0

1

2

3

4

5

f(x)

10

20

30

40

50

60

70

80

90

g(x)

-1

-2

-3

-4

-5

-6

-7

-8

-9

a. (3f+2g(1) b. (fog)(-1)
12. Express the following function, F(x) as a composition of two functions f and g
f(x)= x^2/(x^2+4)
13. You have a coupon for your favorite clothing store for $25 off any purchase of more than $50. The store is also running a 20%-off sale on its entire inventory. Let x be the original price, f(x) be the price with the $25 coupon applied, and g(x) be the price with the 20% discount applied.
a. Write an expression for f(x)
b. Write an expression for g(x)
c. What would the expression (fog)(x) represent?
d. What would the expression (gof)(x) represent?
e. If the store allows you to apply both the 20% discount and the $25-off coupon, does it matter which you apply first? How do you know?

#1 in a linear function, equal spacing in x produces equal spacing in y. The ratio is called the slope, and is constant. Only table #3 meets that condition.

#2
A function whose domain is [2, infinity): √(x-2)
An arithmetic sequence: 1,2,3,4,...
A system of equations with no solutions:
x+y=1
x+y=2

#3
f(x) = 1/(x^2-3), find:
f(3) = 1/(3^2-3) = 1/6
f(2+h) = 1/((2+h)^2-3) = 1/(h^2+4h-1)

#4
f(x) = 1/x-5 and g(x) = x^2+2, find:
1. (fog)(x) = f(g(x)) = 1/(g-5) = 1/(x^2+2-5) = 1/(x^2-3)
2. (gof)(6) = g(f(6)) = g(1) = 1^2+2 = 3

#5
an=4n+5, answer the following:
Find the first five terms: 9,13,17,21,25
Find the sum of the first 25 terms: (25/2)(2*9+24*4) = 1425
Is this an arithmetic sequence? Yes, because A_n+1 - A_n = 4(n+1)+5 - (4n+5) = 4, a constant

Thanks Steve for the help

Sure, I can help you with these questions. Let's go through each one step by step.

1. To determine which table represents a linear function, we need to check if the y-values in the table increase or decrease at a consistent rate.

Table #1: (x, y) = (2, 1), (5, 3), (8, 7), (11, 13), (14, 21), (17, 31), (20, 43)
Table #2: (x, y) = (1, 10), (2, 13), (3, 18), (4, 21), (5, 26), (6, 29), (7, 34)
Table #3: (x, y) = (2, 1), (4, 6), (6, 11), (8, 16), (10, 21), (12, 26), (14, 31)

Looking at the y-values, we can see that in Table #1, the y-values increase by a consistent amount each time (2, 3, 6, 8, 10, 12), indicating a linear function. Therefore, Table #1 represents a linear function.

To find the equation of the linear function, we need to find the slope (m) and the y-intercept (b). The slope can be found by dividing the change in y-values by the change in x-values. Let's take the first two points in Table #1: (2, 1) and (5, 3).

Slope (m) = (3 - 1) / (5 - 2) = 2 / 3

Now, we can use the point-slope form of a linear equation to find the equation of the line. Using the coordinates of the first point, (2, 1), we have:

y - y1 = m(x - x1)
y - 1 = (2/3)(x - 2)

Simplifying the equation, we get:

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

Therefore, the equation of the linear function represented by Table #1 is y = (2/3)x + 2/3.

2. Examples for each of the following:

a. A function whose domain is [2, infinity):
A possible example would be f(x) = √(x - 2), where the square root is defined for values greater than or equal to 2.

b. An arithmetic sequence:
An example of an arithmetic sequence is {2, 4, 6, 8, 10}. In this sequence, the difference between consecutive terms is constant (2), indicating an arithmetic sequence.

c. A system of equations with no solutions:
An example of a system of equations with no solutions is:
x + y = 5
2x - 2y = 7
When trying to solve this system, we will end up with inconsistent equations, showing that there is no common solution.

3. For f(x) = 1 / (x^2 - 3):

a. To find f(3), substitute x = 3 into the function:
f(3) = 1 / (3^2 - 3) = 1 / (9 - 3) = 1 / 6

b. To find f(2+h), substitute x = 2+h into the function:
f(2+h) = 1 / ((2+h)^2 - 3) = 1 / (4 + 4h + h^2 - 3) = 1 / (h^2 + 4h + 1)

4. For f(x) = 1 / (x - 5) and g(x) = x^2 + 2:

a. To find (fog)(x), we need to substitute g(x) into f(x):
(fog)(x) = f(g(x)) = f(x^2 + 2) = 1 / ((x^2 + 2) - 5) = 1 / (x^2 - 3)

b. To find (gof)(6), we need to substitute f(x) into g(x):
(gof)(6) = g(f(6)) = g(1 / (6 - 5)) = g(1) = 1^2 + 2 = 3

5. For the sequence given by a(subscript n) = 4n + 5:

a. To find the first five terms, substitute n = 1, 2, 3, 4, 5 into the sequence:
a1 = 4(1) + 5 = 9
a2 = 4(2) + 5 = 13
a3 = 4(3) + 5 = 17
a4 = 4(4) + 5 = 21
a5 = 4(5) + 5 = 25

The first five terms are: 9, 13, 17, 21, 25.

b. To find the sum of the first 25 terms, we can use the formula for the sum of an arithmetic sequence:
Sn = (n/2)(a1 + an)

In this case, n = 25, a1 = 9, and an = 4n + 5:

S25 = (25/2)(9 + (4(25) + 5)) = (25/2)(9 + 105) = (25/2)(114) = 1425

The sum of the first 25 terms is 1425.

c. This is an arithmetic sequence because the common difference between consecutive terms is constant. In this case, the difference is always 4.

6. To graph the area bound by y < 1/2x + 6, x + 3y >= 12, x >= 0, and x <= 12:

First, graph the lines represented by the inequalities y = 1/2x + 6 and x + 3y = 12.

Start by finding the x and y-intercepts for both lines:

For y = 1/2x + 6:
x-intercept: Set y = 0 => 0 = 1/2x + 6 => x = -12
y-intercept: Set x = 0 => y = 1/2(0) + 6 => y = 6

For x + 3y = 12:
x-intercept: Set y = 0 => x = 12
y-intercept: Set x = 0 => y = 4

Now, plot these points and draw the lines.

Next, shade the region that satisfies the given inequalities. Since y < 1/2x + 6, shade the region below the line y = 1/2x + 6. Also, since x + 3y >= 12, shade the region above the line x + 3y = 12.

Finally, identify the area of overlap between the shaded regions.

7. For the function defined by:
f(x) = { x^2, x <= 1 }
{ 2x + 1, x > 1 }

a. To evaluate f(0), we substitute x = 0 into the function:
f(0) = 0^2 = 0

b. To graph f(x), plot the function for x <= 1 and x > 1 separately.
For x <= 1, plot the points (0, 0) and (1, 1). Connect the points.
For x > 1, plot the function y = 2x + 1 starting from x = 1.

8. To solve the system of equations algebraically:
3x - y = 0
5x + 2y = 22

Multiply the first equation by 2:
6x - 2y = 0

Add the modified first equation to the second equation:
6x - 2y + 5x + 2y = 22
11x = 22
x = 2

Substitute the value of x into the first equation:
3(2) - y = 0
6 - y = 0
y = 6

The solution to the system of equations is x = 2 and y = 6. To verify the solution, you can substitute these values into both equations and check if they hold true.

9. To solve the system of equations algebraically:
8x - 2y = 5
-12x + 3y = 7

Multiply the first equation by 6 and the second equation by 4 to eliminate y:
48x - 12y = 30
-48x + 12y = 28

Add the equations together:
48x - 12y + (-48x + 12y) = 30 + 28
0 = 58

Since 0 does not equal 58, this system of equations has no solution.

10. Given matrices A, B, and C:

A = [ 2 -1 0 ]
[ 0 5 0.3 ]
[ 1 4 10 ]

B = [ 5 0 2 ]
[ 1 -3 9 ]
[ 2 0 4 ]

C = [ 1 3 5 ]

a. To compute 3A + B, multiply each element of matrix A by 3 and then add the corresponding elements of matrix B:
3A + B = [ 6 -2 6 ]
[ 1 12 3.9 ]
[ 5 12 22 ]

b. Matrix addition is only possible when the two matrices have the same dimensions, so it is not possible to add matrix B to matrix C.

c. Matrix multiplication is performed by multiplying each element of the matrix by a constant, so multiplying matrix C by A would result in:
CA = [ 2 -3.9 10.7 ]

11. Given the table:

x -3 -2 -1 0 1 2 3 4 5
f(x) 10 20 30 40 50 60 70 80 90
g(x) -1 -2 -3 -4 -5 -6 -7 -8 -9

a. To evaluate (3f + 2g)(1), substitute x = 1 into each function, and then perform the calculation:
(3f + 2g)(1) = (3 * 10) + (2 * (-1)) = 30 - 2 = 28

b. To find (fog)(-1), first find the value of g(-1) and then substitute it into f(x):
g(-1) = -3
(fog)(-1) = f(g(-1)) = f(-3)

From the table, we can determine that f(-3) = 30.

Therefore, (fog)(-1) = 30.

12. Express the function F(x) as a composition of two functions f and g:
F(x) = f(g(x))

To do this, we need to find two functions, f(x) and g(x), such that when we substitute g(x) into f(x), we get F(x).

In this case, let's take g(x) = x^2 + 4.

And f(x) = x / (x^2 + 4).

Therefore, F(x) = f(g(x)) = (x^2 + 4) / (x^2 + 4).

13. Let's answer the questions related to the clothing store scenario:

a. The expression for f(x), the price with the $25 coupon applied, would be:
f(x) = x - 25

b. The expression for g(x), the price with the 20% discount applied, would be:
g(x) = x - 0.2x = 0.8x

c. The expression (fog)(x) represents applying the 20% discount first and then the $25 coupon:
(fog)(x) = f(g(x)) = (0.8x) - 25

d. The expression (gof)(x) represents applying the $25 coupon first and then the 20% discount:
(gof)(x) = g(f(x)) = 0.8(x - 25) = 0.8x - 20

e. It does not matter which discount is applied first because applying the 20% discount before the $25 coupon or vice versa will give the same final price. This can be seen by evaluating (fog)(x) and (gof)(x) for any value of x example and checking if they are equal.