Choose the correct statement to describe a true relationship between any point on f(x) and -f(x)

Question

If (x,y) is on the graph of f(x), then (-x, - y) is on the graph of -f(x)
If (x,y) is on the graph of f(x), then (x,y) is on the graph of -f(x)
If (x, y) is on the graph of f(x), then (x, -y) is on the graph of -f(x)
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 relationships between the graphs of \( f(x) \) and \( -f(x) \), let's break down each statement:

1. If (x,y) is on the graph of f(x), then (-x, -y) is on the graph of -f(x)
   This statement describes a reflection about the origin, which is not correct for the relationship between \( f(x) \) and \( -f(x) \).

2. If (x,y) is on the graph of f(x), then (x,y) is on the graph of -f(x)
   This is incorrect because if \( (x,y) \) is on the graph of \( f(x) \), it means that \( y = f(x) \), and thus \( (x,y) \) cannot also be on the graph of \( -f(x) \).

3. If (x,y) is on the graph of f(x), then (x, -y) is on the graph of -f(x)
   This statement is correct. If \( (x,y) \) is on the graph of \( f(x) \) (which means \( y = f(x) \)), then \( -y = -f(x) \), which means \( (x, -y) \) is indeed on the graph of \( -f(x) \).

4. If (x,y) is on the graph of f(x) then (-x, y) is on the graph of -f(x)
   This statement is not correct. It does not correctly relate the points on the graph of \( f(x) \) and \( -f(x) \).

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