Question

The range of the function $f(x)=\log _{5}\left(25-x^{2}\right)$ is

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

Answer 1

Answer: (D)

Clearly, $f(x)$ is defined, if
$\Rightarrow 25-x^{2} > 0 $
$ - 5 < x < 5$
Let $y=\log _{5}\left(25-x^{2}\right)$, then
$ 5^{y} =25-x^{2} $
$\Rightarrow x^{2} =25-5^{y}$
$\Rightarrow x =\pm \sqrt{25-5^{y}}$
For $x$ to be real, we must have
$ 25-5^{y} \geq 0 $
$\Rightarrow 5^{y} \leq 25$
$\Rightarrow y \leq 2$
Also, $y=f(x) \rightarrow-\infty$ as $x \rightarrow \pm 5 .$
Hence, range $(f)=(-\infty, 2]$