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:

$(f.g)(x) = 3x^3-3x^2-24x+36$

Domain of (f.g)(x) = All real numbers  

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

Domain of (f/g)(x) = All Real Numbers

Step-by-step explanation:

Given functions are:

$f(x) = x^2+x-6\\g(x) = 3x-6$

We have to calculate f.g and f/g

In order to find f.g we have to multiply both functions

$(f.g)(x) = (x^2+x-6)(3x-6)\\= 3x(x^2+x-6) - 6(x^2+x-6)\\= 3x^3+3x^2-18x-6x^2-6x+36\\= 3x^3+3x^2-6x^2-18x-6x+36\\=3x^3-3x^2-24x+36$

The domain of (f.g)(x) is all real numbers

Now

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

The domain of (f/g)(x) is all real numbers.

Hence,

$(f.g)(x) = 3x^3-3x^2-24x+36$

Domain of (f.g)(x) = All real numbers  

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

Domain of (f/g)(x) = All Real Numbers