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

Question

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 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 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 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 analyze the relationship between any point on the graph of \( f(x) \) and \( -f(-x) \), let's start with the definition of these functions.

1. If \( (x,y) \) is a point on the graph of \( f(x) \), it means that \( y = f(x) \).
2. To find the corresponding point for \( -f(-x) \), we need to evaluate \( -f(-x) \):
   • If we substitute \( -x \) into \( f(x) \), we get \( f(-x) \).
   • Therefore, \( -f(-x) \) would give us \( -f(-x) \).
3. The point that corresponds to \( (x,y) \) under the transformation leading to \( -f(-x) \) would specifically be \( (-x, -y) \) because:
   • We take \( x \) and transform it to \( -x \) and take \( y \) (which is \( f(x) \)) and transform it to \( -y \).

Thus, we can conclude that: If \( (x,y) \) is on the graph of \( f(x) \), then \( (-x, -y) \) is on the graph of \( -f(-x) \).

The correct choice from the response options provided is: If (x,y) is on the graph of f(x), then (−x,−y) is on the graph of −f(−x).