Final answer:
The functions h(x) = x - 6 and f(x) = x^2 - 3x both have 3 in their domain because substituting x = 3 into these functions yields valid outputs. The function g(x) = x / (x - 3) does not have 3 in its domain because it results in division by zero, which is undefined.
Step-by-step explanation:
To determine which functions have 3 in their domain, we need to check if substituting x = 3 into each function yields a valid output. The domain of a function includes all the input values (x-values) for which the function is defined, and we must ensure that no rules for the function's domain are violated when x = 3.
For the function h(x) = x - 6, if we substitute x = 3, we get h(3) = 3 - 6 = -3, which is a valid real number. Therefore, x = 3 is in the domain of h(x).
For the function g(x) = \frac{x}{x - 3}, if we substitute x = 3, we get g(3) = \frac{3}{3 - 3} = \frac{3}{0}, which is undefined because division by zero is not allowed in mathematics. Thus, x = 3 is not in the domain of g(x).
For the function f(x) = x^2 - 3x, substituting x = 3 gives us f(3) = 3^2 - 3(3) = 9 - 9 = 0, which is a valid real number. So, x = 3 is in the domain of f(x).
Therefore, the functions that have 3 in their domain are h(x) and f(x)