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

Question

Responses

If (x,y) is on the graph of f(x), then (x,y) is on the graph of −f(−x).
If left parenthesis x comma y right parenthesis is on the graph of f left parenthesis x right parenthesis , then left parenthesis x comma y right parenthesis is on the graph of negative f left parenthesis negative x right parenthesis .

If (x,y) is on the graph of f(x), then (x,−y) is on the graph of −f(−x).
If left parenthesis x comma y right parenthesis is on the graph of f left parenthesis x right parenthesis , then left parenthesis x comma negative y right parenthesis is on the graph of negative f left parenthesis negative x right parenthesis .

If (x,y) is on the graph of f(x), then (−x,y) is on the graph of −f(−x).
If left parenthesis x comma y right parenthesis is on the graph of f left parenthesis x right parenthesis , then left parenthesis negative x comma y right parenthesis is on the graph of negative f left parenthesis negative x right parenthesis .

If (x,y) is on the graph of f(x), then (−x,−y) is on the graph of −f(−x).
If left parenthesis x comma y right parenthesis is on the graph of f left parenthesis x right parenthesis , then left parenthesis negative x comma negative y right parenthesis is on the graph of negative f left parenthesis negative x right parenthesis .
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Answer 1

To determine the correct relationship between \(f(x)\) and \(-f(-x)\), let's analyze what happens to a point \((x, y)\) on the graph of \(f(x)\):

1. If \((x, y)\) is on the graph of \(f(x)\), this means that \(y = f(x)\).
2. To find the corresponding point for \(-f(-x)\):
   • First, evaluate \(-x\), which reflects the point horizontally across the y-axis. This gives us the function value at \(-x\), which is \(f(-x)\).
   • Next, apply the negative to this value to get \(-f(-x)\). Therefore, the point corresponding to \(-f(-x)\) would be \((-x, -f(-x))\).

So, the statement that correctly describes the relationship would be:

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

Thus, the correct response is:

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