Choose the statement that best describes a true relationship between any point on f(x)

Question

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 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 x comma y right parenthesis is on the graph of negative f left parenthesis negative x right parenthesis .

Answer 1

To understand the relationship between \( f(x) \) and \( -f(-x) \), we analyze what it means for a point \( (x, y) \) to be on the graph of \( f(x) \):

1. If \( (x, y) \) is on the graph of \( f(x) \), that means \( y = f(x) \).

2. The point we want to check is \( (-x, -y) \).

3. Evaluating \( -f(-x) \):

   • First, compute \( -x \) in the function. If \( x \) is replaced with \( -x \) in \( f(x) \), that gives us \( f(-x) \).
   • To find \( -f(-x) \), we take the negative of that value. Thus, \( -f(-x) \) will give us \( -y \) because \( y = f(x) \) and \( f(-x) \) will produce the output corresponding to the input of \( -x \).

Putting this all together, if \( (x, y) \) is on the graph of \( f(x) \), then:

• The point \( (-x, -y) \) will satisfy the equation \( -f(-x) \).

Therefore, the best statement that describes the relationship between any point on \( f(x) \) and \( -f(-x) \) is:

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

So the correct response is:

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