Given functions f and g, perform the indicated operations.
-Find the domain of $\left( \frac { \mathrm { f } } { \mathrm { g } } \right) ( \mathrm { x } )$ when $\mathrm { f } ( \mathrm { x } ) = 8 \mathrm { x } ^ { 2 } - 4 \mathrm { x }$ and $\mathrm { g } ( \mathrm { x } ) = \mathrm { x } ^ { 2 } - 8 \mathrm { x } - 6$.
A) Domain: $( - \infty , 4 - \sqrt { 22 } ) \cup ( 4 - \sqrt { 22 } , 4 + \sqrt { 22 } ) \cup ( 4 + \sqrt { 22 } , \infty )$
B) Domain: $( - \infty , \infty )$
C) Domain: $( - \infty , 4 - \sqrt { 22 } ) \cup ( 4 - \sqrt { 22 } , \infty )$
D) Domain: $( - \infty , 4 - \sqrt { 22 } ) \cap ( 4 - \sqrt { 22 } , 4 + \sqrt { 22 } ) \cap ( 4 - \sqrt { 22 } , \infty )$

Question

Quizplus · Accepted Answer

The domain of $\left( \frac { \mathrm { f } } { \mathrm { g } } \right) ( \mathrm { x } )$ is determined by the values of $x$ for which $g(x) \neq 0$, since division by zero is undefined. Solving $x^2 - 8x - 6 = 0$ for $x$ gives the roots $4 - \sqrt{22}$ and $4 + \sqrt{22}$. Thus, the domain excludes these points but includes all other real numbers, matching option A.

Given Functions F and G, Perform the Indicated Operations (fg)(x)\left( \frac { \mathrm { f } } { \mathrm { g } } \right) ( \mathrm { x } )(gf​)(x)

Given Functions F and G, Perform the Indicated Operations $\left( \frac { \mathrm { f } } { \mathrm { g } } \right) ( \mathrm { x } )$