Find the domain of the function.
-$$f ( x ) = \log _ { 10 } \frac { x + 4 } { x - 4 }$$
A) $( - \infty , - 4 ) \cup ( 4 , \infty )$
B) $( 4 , \infty )$
C) $( - 4,4 )$
D) $( - \infty , - 4 )$

Question

Quizplus · Accepted Answer

For the function to be defined, the argument of the logarithm, $\frac{x+4}{x-4}$, must be positive. This means both the numerator and denominator must be of the same sign. The denominator $x-4$ changes sign at $x=4$, so we consider intervals to the left and right of 4. For $x>4$, both $x+4$ and $x-4$ are positive, making the fraction positive. For $x<4$, $x+4$ can still be positive if $x>-4$, but $x-4$ is negative, making the fraction negative, which is not allowed. Therefore, the domain excludes the interval where $x$ is between -4 and 4, including -4 but excluding 4 itself, because at $x=4$, the denominator becomes 0, leading to division by zero. Hence, the domain is $(-\infty, -4) \cup (4, \infty)$.

Find the Domain of the Function f(x)=log⁡10x+4x−4f ( x ) = \log _ { 10 } \frac { x + 4 } { x - 4 }f(x)=log10​x−4x+4​

Find the Domain of the Function $f ( x ) = \log _ { 10 } \frac { x + 4 } { x - 4 }$