For f(x)=(3x^2)+5and g(x)=7x-2,

a. Verify: g(x + 2) ≠ g(x) + g(2).

b. Find (f – g)(x).

c. Using the resulting function in (b), show that (f – g)(2) = f(2) – g(2).
(The work should be different for each side of the equation.)

d. Is (fg)(0) =(f/g)(0)? Explain.

e. Find (f(x+h)-f(x))/h , h ≠ 0.

The instructions appear quite clear, where does your problem arise?

I will do a)
Left Side = g(x+2) = 7(x+2) - 2
= 7x + 14 - 2 = 7x + 12

Right Side = g(x) + g(2)
= 7x - 2 + 12 - 2
= 7x +8

Thus g(x+2) ≠ g(x) + g(2)

do the others in a similar way

a. To verify if g(x + 2) ≠ g(x) + g(2), we need to substitute x + 2 into the equation g(x) and see if it leads to a different expression than g(x) + g(2).

First, let's find g(x + 2):
g(x + 2) = 7(x + 2) - 2
= 7x + 14 - 2
= 7x + 12

Now, let's find g(x) + g(2):
g(x) + g(2) = (7x - 2) + (7(2) - 2)
= 7x - 2 + 12
= 7x + 10

Since 7x + 12 ≠ 7x + 10, we can conclude that g(x + 2) ≠ g(x) + g(2). Therefore, the verification is true.

b. To find (f - g)(x), we need to subtract g(x) from f(x):

(f - g)(x) = f(x) - g(x)
= (3x^2 + 5) - (7x - 2)
= 3x^2 + 5 - 7x + 2
= 3x^2 - 7x + 7

Therefore, (f - g)(x) is equal to 3x^2 - 7x + 7.

c. Now let's evaluate (f - g)(2) and f(2) - g(2):

(f - g)(2) = 3(2)^2 - 7(2) + 7
= 3(4) - 14 + 7
= 12 - 14 + 7
= 5

f(2) - g(2) = (3(2)^2 + 5) - (7(2) - 2)
= (3(4) + 5) - (14 - 2)
= (12 + 5) - (12)
= 17 - 12
= 5

Both sides of the equation evaluate to the same value, 5. Therefore, (f - g)(2) = f(2) - g(2).

d. To determine if (fg)(0) = (f/g)(0), let's evaluate them separately:

(fg)(0) = f(0) * g(0)
= ((3(0)^2 + 5) * (7(0) - 2)
= (5 * -2)
= -10

(f/g)(0) = f(0) / g(0)
= (3(0)^2 + 5) / (7(0) - 2)
= (5 / -2)
= -2.5

Since -10 ≠ -2.5, (fg)(0) is not equal to (f/g)(0).

e. To find (f(x+h) - f(x)) / h, we'll substitute f(x+h) and f(x) into the formula and simplify:

(f(x+h) - f(x)) / h = [(3(x+h)^2 + 5) - (3x^2 + 5)] / h
= [(3(x^2 + 2xh + h^2) + 5) - (3x^2 + 5)] / h
= (3x^2 + 6xh + 3h^2 + 5 - 3x^2 - 5) / h
= (6xh + 3h^2) / h
= 6x + 3h

Therefore, (f(x+h) - f(x)) / h = 6x + 3h.