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 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 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 .

Answer 1

To analyze the statement involving a point on the graph of \( f(x) \) and its relation to \( -f(-x) \), we start by noting that if \( (x, y) \) is a point on the graph of \( f(x) \), this means:

\[
f(x) = y
\]

We want to look at the point \( -f(-x) \). To find the \( y \)-coordinate of this point, we first evaluate \( -f(-x) \):

1. Substitute \( -x \) into \( f(x) \): If \( (x, y) \) is on \( f(x) \), then \( f(-x) \) gives us the value when the input is \( -x \).
2. The \( y \)-coordinate for the point corresponding to \( -f(-x) \) would thus be:
\[
-f(-x)
\]

Now considering the transformations:
- \( f(-x) \) reflects the graph of \( f(x) \) over the y-axis.
- \( -f(-x) \) reflects that result over the x-axis.

Thus, if \( (x, y) \) is a point on the graph of \( f(x) \), we have the following final relationship based on the transformations applied:
- The transformation gives us \( (-x, -y) \) on the graph of \( -f(-x) \).

Therefore, the statement that best describes the relationship is:

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