Final answer:
A: (-∞, 7) ∪ (7, ∞).The domain of the function f(x) = (3x - 4) / (7 - x) is (-∞, 7) ∪ (7, ∞).
Step-by-step explanation:
The correct answer is option A: (-∞, 7) ∪ (7, ∞).
To find the domain of the function f(x) = (3x - 4) / (7 - x), we need to determine the set of all possible values for x that make the function defined.
We know that the denominator 7 - x cannot be zero, since division by zero is undefined. Therefore, we solve the equation 7 - x ≠ 0 to find the excluded value of x.
7 - x ≠ 0
x ≠ 7
So, the domain of the function is all real numbers except x = 7, which can be written as:
(-∞, 7) ∪ (7, ∞)
To find the domain of the function f(x) = (3x - 4) / (7 - x), you must consider the values of x for which the function is defined. The only restriction comes from the denominator, as it cannot be equal to zero because division by zero is undefined. Therefore, we set the denominator equal to zero and solve for x: 7 - x = 0, hence x = 7. This means x cannot equal 7.
Other than x = 7, there are no restrictions on the values x can take. Therefore, the domain of the function is all real numbers except for 7. In interval notation, this is expressed as the union of all real numbers less than 7 and all real numbers greater than 7.