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

검증된 단계별 안내

Identify the function given: \(f(x) = -\sqrt{-x}\). Notice that the square root is applied to \(-x\), which affects the domain of the function.

Determine the domain by setting the expression inside the square root to be greater than or equal to zero: \(-x \geq 0\). Solve this inequality to find the domain of \(f(x)\).

Rewrite the function to understand its behavior: since \(f(x) = -\sqrt{-x}\), for each \(x\) in the domain, compute \(\sqrt{-x}\) and then take the negative of that value.

Create a table of values by choosing \(x\) values from the domain, calculating \(-x\), then finding \(\sqrt{-x}\), and finally applying the negative sign to get \(f(x)\).

Plot the points from the table on the coordinate plane and sketch the graph, noting that the graph will be a reflection and transformation of the basic square root function due to the negative signs.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 the valid x-values to graph the function.

Square Root Function and Transformations

The square root function, √x, produces non-negative outputs and has a characteristic curve starting at the origin. Transformations such as reflections, shifts, and stretches modify this graph. For example, a negative sign outside the root reflects the graph across the x-axis.

Graphing Functions with Negative Inputs

When the function involves √(-x), the input to the square root is the negation of x, which affects the domain and shape of the graph. This typically reflects the graph of √x across the y-axis, restricting the domain to x ≤ 0. Recognizing this helps in accurately plotting the function.