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
$25-5^y\ge 0$
$\Rightarrow 5^y\le 25$
$\Rightarrow y\le 2$
Also, $y=f(x)\to -\infty$ as $x\to \pm 5.$
Hence, range $(f)=(-\infty ,2]$