Suppose that the functions f and g are defined for all real numbers x as follows.f(x)=x+4,g(x)=3x+2.Write the expressions for(f*g)(x) and (f-g)(x) and evaluate (f+g)(4)

To find the expressions for (f*g)(x) and (f-g)(x), we need to perform the given functions' operations.

1. (f * g)(x) represents the product of f(x) and g(x). To find it, we substitute g(x) into f(x):
(f * g)(x) = f(x) * g(x) = (x + 4) * (3x + 2)
Expanding the expression using the distributive property:
(f * g)(x) = 3x^2 + 2x + 12x + 8
Simplifying further:
(f * g)(x) = 3x^2 + 14x + 8

2. (f - g)(x) represents the difference between f(x) and g(x). So, we subtract g(x) from f(x):
(f - g)(x) = f(x) - g(x) = (x + 4) - (3x + 2)
Expanding the expression:
(f - g)(x) = x + 4 - 3x - 2
Simplifying further:
(f - g)(x) = -2x + 2

Now, let's evaluate (f + g)(4):
To find (f + g)(4), we substitute x = 4 into f(x) and g(x) and then sum their results:
(f + g)(4) = f(4) + g(4) = (4 + 4) + (3*4 + 2)
Calculating within parentheses:
(f + g)(4) = 8 + (12 + 2)
Simplifying further:
(f + g)(4) = 8 + 14
(f + g)(4) = 22

Therefore, the value of (f + g)(4) is 22.

To find the expressions for (f*g)(x) and (f-g)(x), we need to multiply and subtract the corresponding functions respectively.

1. (f*g)(x) = f(x) * g(x)
= (x + 4) * (3x + 2)
= 3x^2 + 2x + 12x + 8
= 3x^2 + 14x + 8

2. (f-g)(x) = f(x) - g(x)
= (x + 4) - (3x + 2)
= x + 4 - 3x - 2
= -2x + 2

Now, let's evaluate (f+g)(4):

3. (f+g)(x) = f(x) + g(x)
= (x + 4) + (3x + 2)
We substitute x = 4:
= (4 + 4) + (3 * 4 + 2)
= 8 + 14
= 22

So, (f+g)(4) = 22.

these are just simple polynomial addition, subtraction and multiplication.

(f*g)(x) = f(x)*g(x) = (x+4)(3x+2)
and the same for the other operators.