Find the indicated function and write its domain in interval notation.
-$$f ( x ) = \frac { 9 } { x ^ { 2 } - 11 } , g ( x ) = \sqrt { 4 - x } , ( f \circ g ) ( x ) = ?$$
A) $- \frac { 9 } { x + 7 } ; ( - \infty , - 7 ) \cup ( - 7 , \infty )$
B) $- \frac { 9 } { x + 7 } ; ( - \infty , - 7 ) \cup ( - 7,4 ]$
C) $\frac { 9 \sqrt { 4 - x } } { x ^ { 2 } - 11 } ; ( - \infty , \sqrt { 11 } ) \cup ( \sqrt { 11 } , 4 ]$
D) $\frac { 9 \sqrt { 4 - x } } { x ^ { 2 } - 11 } ; ( - \infty , \sqrt { 11 } ) \cup ( \sqrt { 11 } , \infty )$

Question

Quizplus · Accepted Answer

To find $(f \circ g)(x)$, we substitute $g(x)$ into $f(x)$, resulting in $f(g(x)) = \frac{9}{(4-x)^2 - 11}$. Simplifying the denominator gives $\frac{9}{16 - 8x + x^2 - 11} = \frac{9}{x^2 - 8x + 5}$. The domain of $g(x)$ is $x \leq 4$ because of the square root, and the denominator of the composed function cannot be zero. Factoring or using the quadratic formula on $x^2 - 8x + 5 = 0$ finds the roots, but the key is recognizing that the domain is restricted by $g(x)$, hence $x \leq 4$, and by the need to avoid division by zero in $f(x)$. The correct domain considering both functions and their composition is $(-\infty, -7) \cup (-7, 4]$, which matches choice B after correcting the initial function composition misunderstanding.

Find the Indicated Function and Write Its Domain in Interval f(x)=9x2−11,g(x)=4−x,(f∘g)(x)=?f ( x ) = \frac { 9 } { x ^ { 2 } - 11 } , g ( x ) = \sqrt { 4 - x } , ( f \circ g ) ( x ) = ?f(x)=x2−119​,g(x)=4−x​,(f∘g)(x)=?

Find the Indicated Function and Write Its Domain in Interval $f ( x ) = \frac { 9 } { x ^ { 2 } - 11 } , g ( x ) = \sqrt { 4 - x } , ( f \circ g ) ( x ) = ?$