Question

Consider the following statements
Statement I The range of the function $f(x)=2-3x,x\in R,x>0$ is $R$.
Statement II The range of the function $f(x)=x^2+2$ is $R$
Coose the correct option.

• A. Statement I is true
• B. Statement II is true
• C. Both are true
• D. Both are false

Answer 1

Answer: (D)

I. We have, $f(x)=2-3x,x\in R,x>0$
Let $f(x)=y$, then
$y=2-3x$
$\Rightarrow 3x=2-y$
$\Rightarrow x=\frac{2-y}{3}$
$\because x>0$
$\Rightarrow \frac{2-y}{3}>0$
$\Rightarrow 2-y>0$
$\Rightarrow 2>y$
$\therefore y<2$
Hence, range of $f=(-\infty ,2)$
$\therefore$ Statement I is false.
II. Now, $f(x)=x^2+2$
Let $y=f(x)$, then
$y=x^2+2$
$\Rightarrow x=\sqrt{y-2}$
$x$ assumes real values, if $y-2\ge 0$
$\Rightarrow y\ge 2$
$\Rightarrow y\in [2,\infty ]$
$\therefore$ Range of $f=[2,\infty ]$