To find $(fg)(x)$, we multiply the functions $f(x)$ and $g(x)$:

$f(x) = x^2 - 5x + 4$ $g(x) = x - 1$

Step 1: Multiply $f(x)$ and $g(x)$

$(fg)(x) = f(x) \cdot g(x)$ $(fg)(x) = (x^2 - 5x + 4)(x - 1)$

Now, use the distributive property to multiply:

$(fg)(x) = x^2(x - 1) - 5x(x - 1) + 4(x - 1)$ $(fg)(x) = x^3 - x^2 - 5x^2 + 5x + 4x - 4$ $(fg)(x) = x^3 - 6x^2 + 9x - 4$

Thus, $(fg)(x) = x^3 - 6x^2 + 9x - 4$

Step 2: Find the domain of $(fg)(x)$

The domain of a product of functions is the set of all values for which both $f(x)$ and $g(x)$ are defined. Since both $f(x)$ and $g(x)$ are polynomials, their domain is all real numbers, except where $g(x) = 0$, because division by zero would be undefined.

$g(x) = x - 1$ Setting $g(x) = 0$, we get: $x - 1 = 0 \quad \Rightarrow \quad x = 1$

Therefore, the domain of $(fg)(x)$ is all real numbers except $x = 1$.

Final Answer:

$(fg)(x) = x^3 - 6x^2 + 9x - 4$ The domain of $(fg)(x)$ is $x \neq 1$.

Would you like more details or explanations?

Here are some related questions for you to consider:

1. How do you factorize $(fg)(x) = x^3 - 6x^2 + 9x - 4$?
2. What is the range of $(fg)(x)$?
3. How do you graph $(fg)(x) = x^3 - 6x^2 + 9x - 4$?
4. Can you find the x-intercepts of $(fg)(x)$?
5. How do you find the critical points of $(fg)(x)$?

Tip: To check the domain of composite or product functions, always check where each function is undefined first.