Question

The range of the function $f\left(x\right)=4-\sqrt{x^2-10x+25}$ is

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

Answer 1

The correct option is A $(-\infty ,4]$

$f\left(x\right)=4-\sqrt{x^2-10x+25}\Rightarrow f\left(x\right)=4-\sqrt{(x-5{)}^2}\Rightarrow f\left(x\right)=4-|x-5|$

We know that, $|x-5|$ lies from $[0,\infty )$
$\therefore -|x-5|$ lies in $(-\infty ,0]$


Adding $4$, we get


Graph will get shifted upwards by $4$ units
$\therefore$ Range of $f\left(x\right)$ is $(-\infty ,4]$