Choose the one alternative that best completes the statement or answers the question.
Write the domain in interval notation.
-$$f ( x ) = \frac { 7 x - 1 } { 4 x ^ { 2 } - 27 x - 81 }$$

A) $\left( - \infty , - \frac { 9 } { 4 } \right) \cup \left( - \frac { 9 } { 4 } , 9 \right) \cup ( 9 , \infty )$
B) $\left( - \infty , - \frac { 9 } { 4 } \right) \cup \left( - \frac { 9 } { 4 } , \frac { 1 } { 7 } \right) \cup \left( \frac { 1 } { 7 } , 9 \right) \cup ( 9 , \infty )$
C) $( - \infty , - 9 ) \cup \left( - 9 , - \frac { 9 } { 4 } \right) \cup \left( \frac { 9 } { 4 } , \infty \right)$
D) $( - \infty , - 9 ) \cup \left( - 9 , - \frac { 1 } { 7 } \right) \cup \left( - \frac { 1 } { 7 } , \frac { 9 } { 4 } \right) \cup \left( \frac { 9 } { 4 } , \infty \right)$

Question

Choose the one alternative that best completes the statement or answers the question.
Write the domain in interval notation.
-$$f ( x ) = \frac { 7 x - 1 } { 4 x ^ { 2 } - 27 x - 81 }$$

A) $\left( - \infty , - \frac { 9 } { 4 } \right) \cup \left( - \frac { 9 } { 4 } , 9 \right) \cup ( 9 , \infty )$
B) $\left( - \infty , - \frac { 9 } { 4 } \right) \cup \left( - \frac { 9 } { 4 } , \frac { 1 } { 7 } \right) \cup \left( \frac { 1 } { 7 } , 9 \right) \cup ( 9 , \infty )$
C) $( - \infty , - 9 ) \cup \left( - 9 , - \frac { 9 } { 4 } \right) \cup \left( \frac { 9 } { 4 } , \infty \right)$
D) $( - \infty , - 9 ) \cup \left( - 9 , - \frac { 1 } { 7 } \right) \cup \left( - \frac { 1 } { 7 } , \frac { 9 } { 4 } \right) \cup \left( \frac { 9 } { 4 } , \infty \right)$

Quizplus · Accepted Answer

The domain of a function is all the values that x can take such that the function is defined. For the given function, the denominator cannot be zero, so we must find the values of x that make $4x^2 - 27x - 81 = 0$. Factoring or using the quadratic formula, we find the roots to be $x = -\frac{9}{4}$ and $x = 9$. These are the points where the function is undefined, so the domain is all real numbers except these two points, which matches option A.

Choose the One Alternative That Best Completes the Statement or Answers