Find the values of x for which the following functions are identical.
$f(x)=\frac{\sqrt{9-x^2}}{\sqrt{x-2}}$ and $g(x)=\sqrt{\frac{9-x^2}{x-2}}$
Answer & explanation
Correct answer: option 2
$f(x)=\frac{\sqrt{9-x^2}}{\sqrt{x-2}}$ is defined if $9-x^2≥0$ and $x-2>0$
$⇒x∈[-3,3]$ and $x>2=x∈(2, 3]$
$g(x)=\sqrt{\frac{9-x^2}{x-2}}$ is defined if $\frac{9-x^2}{x-2}≥0$
$⇒\frac{x^2-9}{x-2}≤0$
From the sign scheme $x ∈ (-∞, -3] ∪ (2, 3]$
Hence, f(x) and g(x) are identical if x = (2, 3]