What is the domain of [tex]f(x)=\frac{3 x}{x-1}[/tex]?
A. all real numbers
B. all nonzero numbers
C. all real numbers except 1
D. all real numbers except 3
Answer (2)
The domain of a function is all possible values of x for which the function is defined.
Identify that f ( x ) = x − 1 3 x is a rational function, which is undefined when the denominator is zero.
Solve x − 1 = 0 to find x = 1 .
Conclude that the domain is all real numbers except 1, so the answer is all real numbers except 1 .
Explanation
Understanding the Problem We want to find the domain of the function f ( x ) = x − 1 3 x . The domain of a function is the set of all possible values of x for which the function is defined.
Rational Functions The given function is a rational function, which means it is defined for all real numbers except where the denominator is equal to zero.
Finding the Undefined Points To find the values of x for which the denominator is zero, we set the denominator equal to zero and solve for x : x − 1 = 0
Solving for x Adding 1 to both sides of the equation, we get: x = 1
Determining the Domain Therefore, the function is undefined when x = 1 . This means the domain of the function is all real numbers except 1.
Final Answer The domain of f ( x ) is all real numbers except x = 1 .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if f ( x ) represents the cost of producing x items, and f ( x ) = x − 1 3 x , then knowing that x cannot be 1 tells us that we cannot produce exactly 1 item using this cost model. Similarly, in physics, if f ( x ) represents the force between two objects at a distance x , and the function is undefined at a certain point, it indicates a singularity or a point where the physical model breaks down. Identifying the domain helps us understand the limitations and applicability of mathematical models in various fields.
The domain of the function f ( x ) = x − 1 3 x includes all real numbers except x = 1 because the function is undefined at this point. Therefore, the correct option is C, 'all real numbers except 1.'
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