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

Answer:

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

Domain: All real numbers

$\frac{f}{g} = \frac{x + 3}{3}$

Domain: x≠2

Step-by-step 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;

$\frac{f}{g} = \frac{f(x)}{g(x)}$

$\frac{f}{g} = \frac{ {x}^{2} + x - 6 }{3x - 6}$

Factor to get:

$\frac{f}{g} = \frac{(x - 2)(x + 3)}{3(x - 2)}$

Cancel common factors:

$\frac{f}{g} = \frac{x + 3}{3}$

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

$x \ne2$