Given functions f and g, perform the indicated operations.
-Find the domain of $( f g ) ( x )$ when $f ( x ) = \sqrt { 5 x + 8 }$ and $g ( x ) = \sqrt { 6 x - 7 }$
A) Domain: $\left[ \frac { 7 } { 6 } , \infty \right)$
B) Domain: $[ 0 , \infty )$
C) Domain: $( - \infty , \infty )$
D) Domain: $\left[ - \frac { 7 } { 6 } , \infty \right]$

Question

Quizplus · Accepted Answer

The domain of $( f g ) ( x )$ requires both $f(x)$ and $g(x)$ to be defined. For $f(x) = \sqrt{5x + 8}$, the inside of the square root must be non-negative: $5x + 8 \geq 0$, which simplifies to $x \geq -\frac{8}{5}$. For $g(x) = \sqrt{6x - 7}$, the inside of the square root must also be non-negative: $6x - 7 \geq 0$, which simplifies to $x \geq \frac{7}{6}$. Since $\frac{7}{6}$ is greater than $-\frac{8}{5}$, the domain of $( f g ) ( x )$ is determined by the more restrictive condition, which is $x \geq \frac{7}{6}$. Therefore, the domain is $\left[ \frac { 7 } { 6 } , \infty \right)$.

Given Functions F and G, Perform the Indicated Operations (fg)(x)( f g ) ( x )(fg)(x)

Given Functions F and G, Perform the Indicated Operations $( f g ) ( x )$