Graph each function. See Examples 1 and 2. ƒ(x) = -√-x

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First, recognize the function given: \(f(x) = -\sqrt{-x}\). This means the square root is taken of the expression \(-x\), and then the result is negated.

Determine the domain of the function. Since the square root function requires the radicand (the expression inside the root) to be non-negative, set \(-x \geq 0\). Solve this inequality to find the domain.

Rewrite the function in terms of \(x\) within the domain. For example, if \(x \leq 0\), then \(-x\) is non-negative, so \(f(x) = -\sqrt{-x}\) is defined there.

Create a table of values by choosing several \(x\) values within the domain, compute \(-x\), then find \(\sqrt{-x}\), and finally apply the negative sign to get \(f(x)\). This will help plot points accurately.

Plot the points on the coordinate plane and sketch the graph. Remember that the graph will be the reflection of the basic square root function \(\sqrt{x}\), first reflected over the y-axis (due to \(-x\) inside the root), and then reflected over the x-axis (due to the negative sign outside the root).

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Domain of a Function

The domain of a function is the set of all input values (x) for which the function is defined. For functions involving square roots, the expression inside the root must be non-negative to yield real values. Understanding the domain helps determine which x-values can be graphed.

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Finding the Domain of an Equation

Square Root Function and Transformations

The square root function, √x, produces outputs that are non-negative and increases slowly as x increases. Transformations such as reflections (negative signs) and shifts affect the graph's shape and position. For example, a negative sign outside the root reflects the graph across the x-axis.

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Domain and Range of Function Transformations

Graphing Functions with Negative Inputs

When the function involves √(-x), the input to the square root is the negation of x, which flips the domain and graph horizontally. This means the function is defined for x ≤ 0, and the graph is a reflection of the standard square root function across the y-axis.

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