$\text { Find } ( f / g ) ( x ) \text {. What is the domain of } f / g \text { ? }$
$f ( x ) = x ^ { 2 } , \quad g ( x ) = 7 x - 5$
A) $- \frac { x ^ { 2 } } { 7 x - 5 } ;$ all real numbers $x$
B) $\frac { 7 x + 5 } { x ^ { 2 } } ;$ all real numbers $x$ except $x = 0$
C) $\frac { x ^ { 2 } } { 7 x - 5 }$; all real numbers $x$ except $x = \frac { 5 } { 7 }$
D) $\frac { 7 x - 5 } { x ^ { 2 } } ;$ all real numbers $x$ except $x = 0$
E) $\frac { x ^ { 2 } } { 7 x + 5 } ;$ all real numbers $x$ except $x = \frac { 7 } { 5 }$

Question

$\text { Find } ( f / g ) ( x ) \text {. What is the domain of } f / g \text { ? }$ 
$f ( x ) = x ^ { 2 } , \quad g ( x ) = 7 x - 5$
A) $- \frac { x ^ { 2 } } { 7 x - 5 } ;$ all real numbers $x$
B) $\frac { 7 x + 5 } { x ^ { 2 } } ;$ all real numbers $x$ except $x = 0$
C) $\frac { x ^ { 2 } } { 7 x - 5 }$; all real numbers $x$ except $x = \frac { 5 } { 7 }$
D) $\frac { 7 x - 5 } { x ^ { 2 } } ;$ all real numbers $x$ except $x = 0$
E) $\frac { x ^ { 2 } } { 7 x + 5 } ;$ all real numbers $x$ except $x = \frac { 7 } { 5 }$

Quizplus · Accepted Answer

To find $(f/g)(x)$, we simply divide $f(x)$ by $g(x)$.

\begin{align*}
\frac{f(x)}{g(x)} &= \frac{x^2}{7x-5} \
&= \frac{x^2}{7x-5} \cdot \frac{\frac{1}{x^2}}{\frac{1}{x^2}} \quad 	ext{(multiply by }\frac{1}{x^2}	ext{ to get a common denominator)}\
&= \frac{1}{7-\frac{5}{x^2}}
\end{align*}

Thus, $(f/g)(x) = \frac{1}{7-\frac{5}{x^2}}$ with a domain of all real numbers $x$ except for $x=0$ and $x=\pm\sqrt{\frac{5}{7}}$.

The only answer choice that matches this is C.

Find (f/g)(x). What is the domain of f/g ? \text { Find } ( f / g ) ( x ) \text {. What is the domain of } f / g \text { ? } Find (f/g)(x). What is the domain of f/g ?

$\text { Find } ( f / g ) ( x ) \text {. What is the domain of } f / g \text { ? }$