Question

Let f(x) = x² + x - 6 and g(x) = 3x - 6, what are f ⋅ g and f/g. state the resulting function and it's domain.

Answer 1

`f \cdot \: g = 3 {x}^(3) - 3 {x}^(2) - 24x + 36`

Domain: All real numbers

`(f)/(g) = (x + 3)/(3)`

Domain: x≠2

Explanation:

The given functions are

`f(x) = {x}^(2) + x - 6`

and

`g(x) = 3x - 6`

`f \cdot \: g = f(x) \cdot \: g(x)`

This implies that:

`f \cdot \: g = (3x - 6)( {x}^(2) + x - 6)`

We expand to get:

`f \cdot \: g = 3x ( {x}^(2) + x - 6) - 6( {x}^(2) + x - 6)`

We expand to get:

`f \cdot \: g = 3 {x}^(3) + 3 {x}^(2) - 18x - 6{x}^(2) - 6 x + 36`

We group like terms to get:

`f \cdot \: g = 3 {x}^(3) - 3 {x}^(2) - 24x + 36`

This is a polynomial function, the domain is all real.

Also;

`(f)/(g) = (f(x))/(g(x))`

`(f)/(g) = \frac{ {x}^(2) + x - 6 }{3x - 6}`

Factor to get:

`(f)/(g) = ((x - 2)(x + 3))/(3(x - 2))`

Cancel common factors:

`(f)/(g) = (x + 3)/(3)`

The domain is all real numbers, except numbers that will make the denominator zero.

`x \\e2`