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 input values for which the function is defined.
Rational functions are undefined when the denominator is zero.
Set the denominator x − 1 equal to zero and solve for x : x − 1 = 0 ⟹ x = 1 .
The domain of f ( x ) = x − 1 3 x is all real numbers except 1: all real numbers except 1 .
Explanation
Understanding the Domain We are asked to find the domain of the function f ( x ) = x − 1 3 x . The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have a rational function, which means it's a fraction where the numerator and denominator are polynomials. Rational functions are defined for all real numbers except where the denominator is equal to zero.
Finding the Restriction To find the values of x that make the denominator 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 This means that the function is undefined when x = 1 . Therefore, 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. The domain helps us understand the limitations and valid inputs for the function in a practical context.
The domain of the function f ( x ) = x − 1 3 x is all real numbers except for 1 . Hence, the correct choice is C. all real numbers except 1.
;
