Question

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 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)
.
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 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)
.
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 .

Answer 1

To determine the correct relationship between any point \((x,y)\) on the graph of \(f(x)\) and the graph of \(f(-x)\), we need to analyze what happens when we evaluate \(f(-x)\).

1. If \((x,y)\) is a point on the graph of \(f(x)\), it means that \(y = f(x)\).
2. To find \(f(-x)\), we would consider the point where \(x\) is replaced by \(-x\). Thus, we evaluate \(f(-x)\) which gives us a new output, say \(k\), where \(k = f(-x)\).

The relationship we are looking for is:
- If \((x, y)\) is on the graph of \(f(x)\), then \((−x, y)\) is on the graph of \(f(−x)\) because when \(x\) is replaced by \(-x\), the output remains \(y\) since \(y = f(x)\) does not change for the original \(x\).

Now, analyzing the answer choices:

- **If (x,y) is on the graph of f(x) then (−x,y) is on the graph of f(−x).** - This is correct because for \(f(-x)\) we simply switch \(x\) with \(-x\) while keeping the same \(y\) value since we are looking at horizontal reflections.

All the other statements imply some sort of transformation that does not hold true for the straightforward relationship between \(f(x)\) and \(f(-x)\).

Therefore, the correct statement is:

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