Question

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

Answer 1

To understand the relationship between \( f(x) \) and \(-f(-x)\), we can analyze it step by step:

1. If \( (x, y) \) is on the graph of \( f(x) \), it means that \( f(x) = y \).

2. To find the point corresponding to \(-f(-x)\):

   • Replace \( x \) with \(-x\) in the function \( f \): this yields \( f(-x) \).
   • Then, taking the negative of that output gives us \(-f(-x)\).

3. Therefore, \(-f(-x) = -y\) if \( f(x) = y \).

So, if we start with the point \((x, y)\) on the graph of \( f(x) \), we will find the corresponding point for \(-f(-x)\) to be \((-x, -y)\).

Thus, the correct statement that describes the relationship is:

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