Find the domain of the composite function f∘g.
-$$f ( x ) = \frac { 8 } { x + 8 } , \quad g ( x ) = \frac { 24 } { x }$$
A) $( - \infty , - 3 )$ or $( - 3,0 )$ or $( 0 , \infty )$
B) $( - \infty , - 8 )$ or $( - 8,0 )$ or $( 0 , \infty )$
C) $( - \infty , - 8 )$ or $( - 8 , - 3 )$ or $( - 3,0 )$ or $( 0 , \infty )$
D) $( - \infty , \infty )$

Question

Quizplus · Accepted Answer

The domain of $g(x) = \frac{24}{x}$ excludes $x = 0$ because division by zero is undefined. For $f(g(x))$, we substitute $g(x)$ into $f(x)$, getting $f(g(x)) = \frac{8}{\frac{24}{x} + 8}$. Simplifying inside the denominator gives $\frac{24}{x} + 8 = \frac{24 + 8x}{x}$. The composite function is undefined when $24 + 8x = 0$, which simplifies to $x = -3$. Therefore, the domain of $f \circ g$ excludes $x = 0$ and $x = -3$, resulting in the intervals $(-\infty, -3)$, $(-3, 0)$, and $(0, \infty)$.

Find the Domain of the Composite Function F∘g f(x)=8x+8,g(x)=24xf ( x ) = \frac { 8 } { x + 8 } , \quad g ( x ) = \frac { 24 } { x }f(x)=x+88​,g(x)=x24​

Find the Domain of the Composite Function F∘g $f ( x ) = \frac { 8 } { x + 8 } , \quad g ( x ) = \frac { 24 } { x }$