Q. Let $f$ be a function with domain $\lbrack - 3,\ 5\rbrack$ and let $g(x) = |3x + 4|$. Then the domain of $(fog)(x)$ is
A) $\left( - 3,\ \frac{1}{3} \right)$
B) $\left\lbrack - 3,\ \frac{1}{3} \right\rbrack$
C) $\left\lbrack - 3,\ \frac{1}{3} \right\rbrack$
D) $\left\lbrack - 3,\ - \frac{1}{3} \right\rbrack$
Explanation
(c)
$(fog)(x) = f\left\lbrack g(x) \right\rbrack = f\left( |3x + 4| \right)$
Since, the domain of $f$ is $\lbrack - 3,\ 5\rbrack$
$\therefore\ \ - 3 \leq |3x + 4| \leq 5$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ |3x + 4| \leq 5$
$\Rightarrow \ \ \ \ - 5 \leq 3x + 4 \leq 5$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ - 9 \leq 3x \leq 1$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ - 3 \leq x \leq \frac{1}{3}$
$\therefore$ Domian of $fog$ is $\left\lbrack - 3,\ \frac{1}{3} \right\rbrack$.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.