Answer
$(f \cdot g)(x) = \frac{2}{x} - 1$
$\text{Domain: all real numbers except } 0$.
Work Step by Step
This exercise asks us to multiply the two functions together. Let's write out the problem:
$(f \cdot g)(x) = f(x) \cdot g(x) = (2 - x)(\frac{1}{x})$
Distribute the terms:
$(f \cdot g)(x) = f(x) \cdot g(x) = (2)(\frac{1}{x}) - (x)(\frac{1}{x})$
Multiply to simplify:
$(f \cdot g)(x) = f(x) \cdot g(x) = \frac{2}{x} - \frac{x}{x}$
Simplify the fractions:
$(f \cdot g)(x) = f(x) \cdot g(x) = \frac{2}{x} - 1$
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 $x$ so that we can exclude them from the domain.
In this exercise, $x$ can be any real number except $0$ since $0$ makes $\frac{2}{x}$ undefined.
