Given Function: f(x) = \(\frac{1}{\sqrt{x+1}}\)

Step 1: Analyze the Square Root and Denominator

\(x + 1 > 0\) | The expression inside the square root must be greater than zero, and the denominator must not be zero.

\(x > -1\) | Subtract 1 from both sides to solve for x.

Summary: The domain of the function \(\frac{1}{\sqrt{x+1}}\) is \(x \in (-1, \infty)\). This means that the function is defined for all real numbers greater than -1.

Given Function: f(x) = 1/√(x+1)

Domain Analysis: The domain is all real numbers greater than -1.

Interval Notation: (-1, ∞)

“(-1” | The parenthesis means that -1 is not included in the domain.

“∞)” | The parenthesis means that the domain extends to positive infinity, without a specific upper bound.

The comma separates the lower and upper bounds of the interval.

Set-Builder Notation: {x | x > -1}

“{” | The curly brace starts the set definition.

“x” | The variable representing the elements in the set.

“|” | The vertical bar means “such that.”

“x > -1” | The condition that the elements in the set must satisfy.

“}” | The curly brace ends the set definition.

Summary: The domain of the function 1/√(x+1) is represented by the interval notation (-1, ∞) and the set-builder notation {x | x > -1}. Both notations describe the set of all real numbers greater than -1.