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Choose the statement that best describes a true relationship between any point on f(x) and - f (-x).

• If (x, y) is on the graph of f(x), then (x, y) is on the graph of - f (-x).

• If (x, y) is on the graph of f (2x), then (x, -y) is on the graph of - f (-x)

• If (x, y) is on the graph of f(x), then (-x, —y) is on the graph of
一f (一x).

• If (x, y) is on the graph of f(x), then (-x, y) is on the graph of — f (-х)

1 answer

To analyze the transformation related to the function \( f(x) \) and the expression \( -f(-x) \), let's break down what this expression represents.

  1. Transformation of \( f(x) \):
    • The \(-x\) in \( f(-x) \) indicates a reflection across the y-axis.
    • The \(-f(-x)\) indicates that after reflecting \( f(x) \) across the y-axis (to get \( f(-x) \)), we reflect that result across the x-axis.

Putting these together, if you start with a point \( (x, y) \) on the graph of \( f(x) \), you can determine the corresponding point on the graph of \( -f(-x) \):

  • Reflecting \( (x, y) \) across the y-axis gives \( (-x, y) \).
  • Then reflecting \( (-x, y) \) across the x-axis gives \( (-x, -y) \).

Thus, if \( (x, y) \) is on \( f(x) \), the corresponding point on \( -f(-x) \) will be \( (-x, -y) \).

So the statement that best describes the true relationship is:

• If (x, y) is on the graph of f(x), then (-x, -y) is on the graph of -f(-x).