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Evaluating an expression means finding a specific value for the variable in a function by replacing it with a particular value.
To evaluate the given expression for the function \( f(x) = -\sqrt{-x} \), you're tasked with substituting \( x \) with \(-100\).
Here's how it’s done:

• Replace \( x \) with \(-100\): This means you substitute every \( x \) in the function with \(-100\).
• Simplify the expression: Take the square root of 100, which is 10, and apply the negative sign in front, resulting in \(-10\).

Therefore, \( f(-100) = -10 \).
This outcome is obtained by following the steps of substitution and simplification, key elements in the evaluation process.

The domain of a function is the set of all possible input values (\( x \)) that the function can accept without causing any mathematical problems.
For the function \( f(x) = -\sqrt{-x} \), special attention is required due to the square root, which is only valid for non-negative numbers.
Since the function involves \(-x\), we need to ensure that \(-x \geq 0\). Simplifying this inequality leads to \( x \leq 0 \).
This shows that \( x \) can be any real number as long as it is less than or equal to zero.
Thus, the domain of this function is expressed in interval notation as \((-\infty, 0]\).
This means the function takes all values on the number line from negative infinity up to and including zero.

Understanding the range of a function involves finding all possible output values (function values) after substituting all values from the domain.
In the case of \( f(x) = -\sqrt{-x} \), the output depends on how \( -x \) results in square roots of positive numbers.
Here's how you can determine the range:

• The expression \(-\sqrt{-x}\) means the square root is always non-negative because square roots yield non-negative results.
• However, there is a negative sign in front of the square root which inversely makes all outputs non-positive.
• When \( x = 0 \), the output is zero: \( f(0) = 0 \).
• For more negative values of \( x \), \( f(x) \) will result in more negative numbers.

Therefore, the range is \((-\infty, 0]\).
This indicates the function produces output values that start from zero and extend all the way to negative infinity.