What is the domain of the function \( f(x) = \frac{1}{x-2} \)?

What is the domain of the function \( f(x) = \frac{1}{x-2} \)?

• A. All real numbers
• B. \( x \neq 0 \)
• C. \( x \neq 1 \)
• D. \( x \neq 2 \)

Answer: (D)

The domain of the function \( f(x) = \frac{1}{x-2} \) consists of all the values of \( x \) for which the function is defined. In this case, the function involves a denominator, \( x - 2 \). A rational function is undefined whenever its denominator equals zero, as division by zero is not allowed in mathematics. To find the value that makes the function undefined, set the denominator equal to zero: \[ x - 2 = 0 \] Solving this gives: \[ x = 2 \] This means that the function \( f(x) \) cannot take the value \( x = 2 \). Therefore, the domain of this function includes all real numbers except \( x = 2 \). Thus, the correct answer specifies that \( x \) must not be equal to 2, which illustrates that the function is well-defined for all other real numbers.

The domain of the function ( f(x) = \frac{1}{x-2} ) consists of all the values of ( x ) for which the function is defined. In this case, the function involves a denominator, ( x - 2 ). A rational function is undefined whenever its denominator equals zero, as division by zero is not allowed in mathematics.

To find the value that makes the function undefined, set the denominator equal to zero:

[

x - 2 = 0

]

Solving this gives:

[

x = 2

]

This means that the function ( f(x) ) cannot take the value ( x = 2 ). Therefore, the domain of this function includes all real numbers except ( x = 2 ).

Thus, the correct answer specifies that ( x ) must not be equal to 2, which illustrates that the function is well-defined for all other real numbers.