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 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)
.

Answer 1

To find the true relationship between any point on \( f(x) \) and \( -f(-x) \), we need to analyze the transformations that occur.

1. If \( (x, y) \) is on the graph of \( f(x) \), then \( y = f(x) \).
2. To get the point for \( -f(-x) \):
- First, replace \( x \) with \( -x \), resulting in \( f(-x) \).
- Then negate this value, leading to \( -f(-x) \), meaning the output becomes \( -f(-x) \).

Now, let’s define what that means for \( y \):
- If \( (x, y) \) is on the graph of \( f(x) \), then:
- \( y = f(x) \),
- The corresponding point for \( -f(-x) \) would be \( (-x, -y) \) because when \( x \) changes to \( -x \), the function value becomes \( f(-x) \) and negating that gives \( -f(-x) \).

Thus, if \( (x, y) \) is on the graph of \( f(x) \), then the point \( (-x, -y) \) will indeed be on the graph of \( -f(-x) \).

The correct statement is:
"If \( (x,y) \) is on the graph of \( f(x) \), then \( (-x,-y) \) is on the graph of \( -f(-x) \)."