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A square root function is a type of algebraic function that involves the square root of a variable expression, like \( \sqrt{x+1} \) in our example. The operation of square rooting is only applicable to non-negative values. This means when we have a function that includes a square root, it places restrictions on the domain - the set of all possible input values (x-values) for the function.

In the function \( g(x)=\sqrt{x+1}-\frac{1}{x} \), the expression inside the square root \( x+1 \) must be greater than or equal to zero. This is because the square root of a negative number is not real, and we only consider real numbers in most algebraic contexts.

So, when solving for the domain concerning the square root, we set:

• \( x+1 \geq 0 \)
• Simplifying, we find \( x \geq -1 \)

This constraint limits the possible x-values starting from -1 and going upwards towards infinity.

A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. In our function, the term \( \frac{1}{x} \) is a rational expression.

Rational expressions are sensitive to values that make the denominator zero, because division by zero is undefined in mathematics. Therefore, it's crucial to identify any x-values that could lead to a zero in the denominator and exclude them from the function's domain.

For \( \frac{1}{x} \), the concern is straightforward:

• The denominator is \( x \)
• \( x \) must not be zero

So, we set the condition \( x eq 0 \), meaning zero must be excluded from the possible x-values. Combining this with other domain restrictions will ensure precise function definitions.

Interval notation offers a concise way to represent the domain of a function. It uses brackets and parentheses to depict the range of values a function's input (x-values) can encompass.

In interval notation:

• Brackets \([ \text{or} ]\) indicate that the endpoint value is included in the domain. For example, \([-1, 0)\) means -1 is included.
• Parentheses \(( \text{or} )\) show that the endpoint is not included. For instance, \((0, \infty)\) means 0 is not included, but we approach infinity without end.

For the function \( g(x)=\sqrt{x+1}-\frac{1}{x} \), the solved domain is

• \([-1, 0) \cup (0, \infty)\)

This expression means x can take values starting from -1 up to but not including 0, and from just above 0 indefinitely into positive infinity. The \( \cup \) symbolizes the union of intervals, merging possible values from both intervals into a complete set.