find the domain \( f(x)=\frac{x+1}{\sqrt{1-\sqrt{x-1}}} \)

To find the domain of the function \( f(x)=\frac{x+1}{\sqrt{1-\sqrt{x-1}}} \), we need to consider the restrictions on the values of \( x \) that make the function defined. 1. The denominator of the function cannot be equal to zero, as division by zero is undefined. Therefore, we need to find the values of \( x \) that make the denominator non-zero. 2. The expression inside the square root, \( 1-\sqrt{x-1} \), must be greater than or equal to zero because the square root of a negative number is not a real number. Let's solve these conditions step by step: 1. Find the values of \( x \) that make the denominator non-zero: \[ 1-\sqrt{x-1} \neq 0 \] \[ \sqrt{x-1} \neq 1 \] \[ x-1 \neq 1 \] \[ x \neq 2 \] 2. Find the values of \( x \) that make the expression inside the square root greater than or equal to zero: \[ 1-\sqrt{x-1} \geq 0 \] \[ \sqrt{x-1} \leq 1 \] \[ x-1 \leq 1 \] \[ x \leq 2 \] Combining the results from both conditions, the domain of the function \( f(x)=\frac{x+1}{\sqrt{1-\sqrt{x-1}}} \) is \( x \in (-\infty, 2] \).