Question

The range of the function $f\left(x\right)=x^2+2x+2$ is

• A. $\left(1,\infty \right)$
• B. $(2,\infty )$
• C. $(0,\infty )$
• D. $[1,\infty )$

Answer 1

The correct option is D

$[1,\infty )$

Find the range of the given function

Given, $f\left(x\right)=x^2+2x+2$

Let, $f\left(x\right)=y$

$$\begin{gathered} \Rightarrow y=x^2+2x+1+1 \\ \Rightarrow y={(x+1)}^2+1 \\ \Rightarrow y-1={(x+1)}^2 \\ \Rightarrow x=\sqrt{y-1}-1 \end{gathered}$$

So, the expression under the square root $\ge 0$

$$\begin{gathered} \Rightarrow y-1\ge 0 \\ \Rightarrow y\ge 1 \end{gathered}$$

So, the range is $[1,\infty )$

Hence, the required answer is $D$, $[1,\infty )$.