Question

The domain of $f\left(x\right)=\frac{1}{\sqrt{2x-1}}-\sqrt{1-x^2}$ is

• A. $]0.5,1]$
• B. $[-1,\infty [$
• C. $[1,\infty [$
• D. None of these

Answer 1

The correct option is A

$]0.5,1]$

Explanation for the correct answer:

Domain of the function:

$f\left(x\right)=\frac{1}{\sqrt{2x-1}}-\sqrt{1-x^2}$

Let $f\left(x\right)=p\left(x\right)-q\left(x\right)$

First, we find the domain of $p\left(x\right)=\frac{1}{\sqrt{2x-1}}$

This is defined when

$$\begin{gathered} 2x-1>0 \\ \Rightarrow x>0.5\left(1\right) \end{gathered}$$

Now we find the domain of $q\left(x\right)=\sqrt{1-x^2}$

This is defined when

$$\begin{gathered} 1-x^2\ge 0 \\ \Rightarrow x^2\le 1 \\ \Rightarrow -1\le x\le 1\left(2\right) \end{gathered}$$

Intersection of $\left(1\right)$and $\left(2\right)$ is the domain of $f\left(x\right)$

$$\begin{gathered} \left(0.5,\infty \right)\cap \left[-1,1\right] \\ =]0.5,1] \end{gathered}$$

Hence, option (A) is the correct option