Question

Choose the statement that best describes a true relationship between any point on f(x) and −f(−x).(1 point)

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(x), 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 −f(−x)

If (x,y) is on the graph of f(x), then (x,y) is on the graph of −f(−x)

Answer 1

To analyze the relationship between a point \((x, y)\) on the graph of \(f(x)\) and the point on the graph of \(-f(-x)\), we need to define what \(-f(-x)\) means.

1. If \((x, y)\) is a point on the graph of \(f(x)\), this means \(y = f(x)\).
2. To find the corresponding point for \(-f(-x)\):
- First, we evaluate \(-f(-x)\):
- Substitute \(-x\) for \(x\) in \(f(x)\): we have \(f(-x)\).
- Now, take the negative: we have \(-f(-x)\).
3. Thus, the value of \(-f(-x)\) corresponds to a new point. If we find this point:
- The input \(-x\) gives us \(y' = f(-x)\), so \(-f(-x) = -y'\).
- We have the point \((-x, -f(-x))\).

So, if \((x, y)\) is on the graph of \(f(x)\), then the point for \(-f(-x)\) must be \((-x, -y)\).

Therefore, the correct statement is:

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