Question

The range of the function $f\left(x\right)={\log }_5(25-x^2)$ is

• A. $[0,5]$
• B. $[0,2)$
• C. $(0,2)$
• D. None of these

Answer 1

The correct option is D None of these

Given, $f\left(x\right)={\log }_5(25-x^2)$

$y={\log }_5(25-x^2)$

$\Rightarrow 5^y=25-x^2$

$\Rightarrow x^2=25-5^y$

$\Rightarrow x=\sqrt{25-5^y}$

For the function in $y$ to be defined

$25-5^y\ge 0$

$\Rightarrow 5^y\le 25$

$\Rightarrow 5^y\le 5^2$

$\Rightarrow y\le 2$

$\Rightarrow y\in (-\infty ,2]$

So, range of the function is $(-\infty ,2]$.

So, option D is correct.