Answer
$(f \cdot g)(x) = 2x^3 - x^2 - 4x + 3$
$\text{The domain is the set of all real numbers.}$
Work Step by Step
This exercise asks us to subtract $f(x)$ from $g(x)$. Let's write out the problem:
$(f \cdot g)(x) = f(x) \cdot g(x) = (2x^2 + x - 3)(x - 1)$
Distribute terms first to get rid of the parentheses, paying attention to the signs:
$(f \cdot g)(x) = f(x) \cdot g(x) = (2x^2)(x) + (2x^2)(-1) + (x)(x) + (x)(-1) + (-3)(x) + (-3)(-1)$
Multiply out the terms:
$(f \cdot g)(x) = f(x) \cdot g(x) = (2x^3) + (- 2x^2) + (x^2) + (-x) + (-3x) + (3)$
Combine like terms:
$(f \cdot g)(x) = f(x) • g(x) = 2x^3 - x^2 - 4x + 3$
When we find the domain, we want to find which values of $x$ will cause the function to become undefined; in other words, we want to find any restrictions for $x$. In this exercise, $x$ can be any real number since there are no restrictions, so the domain is all real numbers.
