Question

Let $f(x)=x^2+4x-1$ and $g(x)=|x|$. If $h(x)=f(g(x))+10$, then the range of $h(x)$, is

• A. $\{y:y\ge 10\}$
• B. $\{y:y\ge 0\}$
• C. $\{y:y\ge 5\}$
• D. $\{y:y\ge 9\}$

Answer 1

Answer: (D)

$h(x)=f(g(x))+10=|x{|}^2+4|x|-1+10=|x{|}^2+4|x|+9=(|x|+2{)}^2+5$
Hence range of $h(x)$ is $[9,\infty )$
Minimum value occurs when $x=0$