Question

If $f:R\to S$ defined by
$f\left(x\right)=4\sin x-3\cos x+1$ is onto, then $S$ is equal to

• A. $[-5,5]$
• B. $(-5,5)$
• C. $(-4,6)$
• D. $[-4,6]$

Answer 1

The correct option is D $[-4,6]$

We have $f\left(x\right)=4sinx-3cosx+1$

Therefore

maximum of $f=\sqrt{4^2+(-3{)}^2}+1=\sqrt{16+9}+1=\sqrt{25}+1=5+1=6$ and

minimum of $f=-\sqrt{4^2+(-3{)}^2}+1=-\sqrt{16+9}+1=-\sqrt{25}+1=-5+1=-4$

$S=$ Range of $f$=$[minimumf,maximumf]=[-4,6]$