Question

The domain and range of the function $f\left(x\right)=\sqrt{{}^{x^2+4x}{C}_{2x^2+3}}$ are

• A. domain: $1,2,3$, range: $1,2\sqrt{3}$
• B. domain: $1,2\sqrt{3}$ range: $1,2,3$
• C. domain: $1,3$ range: $1$
• D. none of these

Answer 1

The correct option is A domain: $1,2,3$, range: $1,2\sqrt{3}$

$f\left(x\right)=\sqrt{{}^{x^2+4x}{C}_{2x^2+3}}$
For this function to exist $x^2+4x\ge 0\Rightarrow x\in R-[-4,0]$
and $x^2+4x\ge 2x^2+3\Rightarrow x^2-4x+3\le 0\Rightarrow (x-3)(x-1)\le 0\Rightarrow x=1,2,3$
Hence domain is $1,2,3$ and range is $\sqrt{{}^5{C}_5},\sqrt{{}^{13}{C}_{11}},\sqrt{{}^{21}{C}_{21}}=1,2\sqrt{3}$