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When we talk about square root functions, we are dealing with functions that involve the square root symbol (√). These functions will produce outputs by extracting the square root of a given input, usually a non-negative number.
In mathematics, we commonly see the square root function written as \(\sqrt{x}\). It simplifies to the value that, when multiplied by itself, gives back the original number \(x\). For example, \(\sqrt{4} = 2\) because \(2 \times 2 = 4\).
However, situations change when the expression involves more complexity, such as negative signs or other operations within the square root. In the context of the problem we are discussing, we have \(-\sqrt{-x}\). This differs from the simple square root of \(x\) by involving a negative input, allowing negative values in a very particular manner.

• This function, \(-\sqrt{-x}\), accepts non-positive inputs.
• The negative sign outside the square root affects the output, flipping the sign of the result.

Understanding these nuances is key to working reliably with square root functions in different forms. Feel free to try substituting various numbers to see how it changes the output! Remember always to handle the expression inside the square root carefully to know what values are permissible for input.

Function evaluation is a simple but crucial concept in math. It involves replacing a variable within a function with a given number and then simplifying it to find the result.
Consider the function \(f(x) = -\sqrt{-x}\). To find \(f(-100)\), you substitute \(x = -100\) directly into the function.
Here's how it works step-by-step:

• First, replace \(x\) with \(-100\): \(-\sqrt{-(-100)}\).
• Then, simplify the term inside the square root: \(-(-100) = 100\).
• Finally, compute the square root and apply the negative sign: \(-\sqrt{100} = -10\).

Thus, when \(x = -100\), the function evaluates to \(-10\). Function evaluation helps us understand what output we can expect from giving specific inputs. It's useful across many areas of math, including finding particular characteristics like the domain and range of functions.

Negative numbers present interesting challenges and features in mathematical functions. Specifically, in our context with \(-\sqrt{-x}\), understanding how negative numbers affect calculations is essential.
Consider the expression \(-\sqrt{-x}\). Here, two negatives interact:

• The negative sign inside the square root ensures that \(x\) must be non-positive (zero or less).
• The negative sign outside the square root then reverses the sign of the output of \(\sqrt{-x}\) after it's been solved.

Negative numbers by themselves imply the opposite of their positive counterparts. However, when employed in mathematical operations like our square root scenario, they enable different solutions and results.
Therefore, working with negative numbers inside functions can dramatically alter what inputs are valid, impacting both the domain and range of the function. It's important always to consider the location of the negative signs in the expression to predict the behavior of the function accurately.