To find the product function $(f \cdot g)(x)$ for $f(x) = x - 5$ and $g(x) = x^2 - 2x + 3$, substitute the expressions into the formula $(f \cdot g)(x) = f(x) \cdot g(x)$. Multiplying the polynomials gives $(x - 5)(x^2 - 2x + 3)$. Distribute the terms to obtain $x^3 - 2x^2 + 3x - 5x^2 + 10x - 15$. Combining like terms yields the simplified product function: $(f \cdot g)(x) = x^3 - 7x^2 + 13x - 15$.