If f(x) = 3x^2 - x-5 and g(x) = x^2 - 8

Question

a. What is (f-g)(x)?

b. Evaluate (f*g)(2)

Answer 1

(f-g)(x) = f(x) - g(x) = 3x^2 - x - 5 - (x^2 - 8) = 2x^2 - x + 3

(f*g)(x) = f(x) * g(x) = (3x^2 -x -5) * (x^2 - 8) = 4x^4 - x^3 - 5x^2 - 24x^2 + 8x + 40 = 4x^4 - x^3 - 29x^2 + 8x + 40

Answer 2

If f(x) = 3x^2 - x-5 and g(x) = x^2 - 8

Then

(f-g)(x) = f(x) - g(x) = 3x^2 - x-5 -(x^2 - 8) = 3x^2 - x-5 -x^2 + 8= 3x^2-x^2 - x + 8 -5

= 2x^2 - x + 3

and

(f*g)(x) = (3x^2 - x-5)(x^2 - 8).

Therefore

(f*g)(2) = (3(2)^2 - 2-5)(2^2 - 8)

= (12 - 2 - 5)(4 - 8) = 5*(-4)

=-20

Answer 3

First we start with the definition of our functions:

The notation is defined to mean . Substituting the definitions of and :

Distributing the negative sign through the right set of parentheses:

Collecting like terms:

Simplifying:

The notation is defined to mean . Substituting the definitions of and :

Distributing \left(3x^2-x-5\right) through \left(x^2-8\right):

Expanding:

Collecting like terms:

Simplifying: