To analyze the relationship between \( f(x) \) and \( -f(-x) \), we need to understand what each transformation entails.
If we have a point \( (x, y) \) on the graph of \( f(x) \), this means \( y = f(x) \).
Now let's examine the point on the graph of \( -f(-x) \):
- Firstly, we compute \( -f(-x) \) at \( x \):
- The input is \( -x \) and hence, the output will be \( -f(-x) \).
- If we evaluate \( f \) at \( -x \), we have:
- \( f(-x) \) gives us some value, say \( z \). Therefore, \( -f(-x) = -z \).
Thus, the corresponding point on the graph of \( -f(-x) \) is \( (-x, -y) \) because:
- When \( x \) is replaced by \( -x \), the point \( (x, y) \) transforms into \( (-x, -y) \) where \( y = f(x) \) and hence, \( -y = -f(-x) \).
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).