Question

Choose the correct statement to describe 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 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 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 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 determine the correct relationship between points on the graphs of \( f(x) \) and \( f(-x) \), let's analyze what it means for a point \( (x, y) \) to lie on the graph of \( f(x) \). This means that \( y = f(x) \).

Now, for \( f(-x) \), we need to find the corresponding point when we input \( -x \) into the function. The point on the graph of \( f(-x) \) would be \( (-x, f(-x)) \).

Thus, if \( (x, y) \) is on the graph of \( f(x) \), then:
- We have \( y = f(x) \).
- The point corresponding to \( -x \) would be \( (-x, f(-x)) \).

So, we have:
- \( (x, y) \) is on the graph of \( f(x) \) ⇒ \( (x, f(x)) \)
- The corresponding point on the graph of \( f(-x) \) would need to evaluate \( -x \), giving us \( (−x, f(−x)) \).

However, none of the provided options exactly describes this relationship correctly. Let's evaluate the options to see which one is factually close:

**Correct Option:**
- **If (x,y) is on the graph of f(x) then (−x,y) is on the graph of f(−x).**

This statement implies that for every point \( (x, f(x)) \) on the graph of \( f(x) \), there is a point \( (-x, f(-x)) \) on the graph of \( f(-x) \), as the value of \( y \) remains the same but the \( x \) value is inverted.

Thus, the correct statement providing the true relationship is:
- **If (x,y) is on the graph of f(x), then (−x,y) is on the graph of f(−x).**