To find the value of \( x \) that must be excluded from the domain of \( (fg)(x) \), we first need to understand how to find \( (fg)(x) \).
The expression \( (fg)(x) \) means \( f(g(x)) \), where:
- \( f(x) = x^2 - 5x + 6 \)
- \( g(x) = x - 2 \)
First, we should determine \( g(x) \): \[ g(x) = x - 2 \] This function is defined for all real numbers, so there are no exclusions here.
Next, we substitute \( g(x) \) into \( f(x) \): \[ (fg)(x) = f(g(x)) = f(x - 2) \]
Now, we need to calculate \( f(x - 2) \): \[ f(x - 2) = (x - 2)^2 - 5(x - 2) + 6 \] Expanding this: \[ = (x^2 - 4x + 4) - (5x - 10) + 6 = x^2 - 4x + 4 - 5x + 10 + 6 = x^2 - 9x + 20 \]
Now, we must look at the function \( g(x) \), which was \( x - 2 \). The only restriction on the domain of \( (fg)(x) \) comes from \( g(x) \) when it leads to a division by zero or other issues. In this case, there is no division, but we should find values of \( x \) where \( g(x) \) is equal to zero because that affects the output of \( (fg)(x) \).
Set \( g(x) = 0 \): \[ x - 2 = 0 \implies x = 2 \]
Thus, the value of \( x \) that must be excluded from the domain of \( (fg)(x) \) is:
\[ \boxed{2} \]