Choose the one alternative that best completes the statement or answers the question.
Find the domain of the composite function f ° g.
-$$f ( x ) = \frac { x } { x + 7 } ; \quad g ( x ) = \frac { 7 } { x + 5 }$$
A) $\{ x \mid x \neq - 5 , x \neq - 6 \}$
B) $\{ x \mid x \neq 0 , x \neq - 5 , x \neq - 6 \}$
C) $\{ x \mid x$ is any real number $\}$
D) $\{ x \mid x \neq - 5 , x \neq - 7 \}$

Question

Quizplus · Accepted Answer

The domain of $g(x) = \frac{7}{x+5}$ excludes $x = -5$ to avoid division by zero. When composing $f(g(x))$, we substitute $g(x)$ into $f$, resulting in $f(g(x)) = \frac{\frac{7}{x+5}}{\frac{7}{x+5} + 7}$. Simplifying the denominator to avoid division by zero, we find that $x \neq -7$ from the original $f(x)$ is not directly relevant because the composition changes the denominator. However, we must ensure the denominator of $g(x)$, $x+5$, does not equal zero, and similarly, the resulting expression from the composition does not introduce any new restrictions beyond what is already stated. Therefore, the correct domain is $x \neq -5$, and we must also consider the effect of the composition on the domain of $f(x)$, which does not introduce new restrictions beyond $x \neq -5$. However, upon closer inspection, the composition $f(g(x))$ indirectly introduces a condition that excludes $x = -6$ because when $x = -6$, the denominator of the composite function becomes zero, which is not allowed. Thus, the correct domain is $x \neq -5, x \neq -6$, making option A correct.

Choose the One Alternative That Best Completes the Statement or Answers