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Chapter 1: Problem 82

Determine whether the statement is true or false. Justify your answer. The graph of \(y=-f(x)\) is a reflection of the graph of \(y=f(x)\) in the \(y\) -axis.

Short Answer

Expert verified
The statement is false. The graph of \(y=-f(x)\) is a reflection of the graph of \(y=f(x)\) across the \(x\)-axis, not the \(y\)-axis.

Step by step solution

01

Understanding Transformations

Replacing a function \(f(x)\) with \(-f(x)\) does produce a reflection, but it's crucial to understand about which axis. This transformation replaces each output value \(f(x)\) with its opposite, thus preserving the \(x\)-value while flipping the \(y\)-value's sign.
02

Identify the Axis of Reflection

To identify the correct axis of reflection, understanding the function's change is essential. Since the \(y\)-values (outputs) are those getting reversed while the \(x\)-valuess remain unchanged, the function does reflect, but along the \(x\)-axis, not the \(y\)-axis.
03

Making the Conclusion

Therefore, the original statement which claims the function reflects along the \(y\)-axis is false. The transformation of \(f(x)\) to \(-f(x)\) results in a reflection about the \(x\)-axis.

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