Question

The table below represents the function f(x)
.
X | -5 -3 -1 0 1 3
F | 7 3 -2 1 -2 3


f(x)
is even.

Which of the following is a true statement? ​​​​​​

(1 point)
Responses

The graph of f(x)
is symmetric about the x-axis.
The graph of f of x is symmetric about the x-axis.

The graph of f(x)
is symmetric about the y-axis.
The graph of f of x is symmetric about the y-axis.

The graph of f(x)
is symmetric about the origin.

Answer 1

To determine the symmetry of the function \( f(x) \) based on the information that it is even, we need to recall the definition of even functions:

A function \( f(x) \) is called even if for every \( x \) in the domain of \( f \):

\[
f(-x) = f(x)
\]

This means that the graph of the function is symmetric about the y-axis.

Given the provided values for \( f(x) \):

- \( f(-5) = 7 \) and \( f(5) = ? \) (not provided)
- \( f(-3) = 3 \) and \( f(3) = 3 \)
- \( f(-1) = -2 \) and \( f(1) = -2 \)
- \( f(0) = 1 \)

We see that:

- \( f(-3) = f(3) \)
- \( f(-1) = f(1) \)
- \( f(0) = f(0) \) (which is trivially true)

Since we have pairs of values that reflect the property of even functions, we can conclude that:

**The graph of \( f(x) \) is symmetric about the y-axis.**

Now let's analyze the options provided in your question:

1. The graph of f(x) is symmetric about the x-axis. (This is false because for a function to be symmetric about the x-axis, it would need to be undefined for negative values of f.)
2. The graph of \( f(x) \) is symmetric about the y-axis. (This is true — \( f(x) \) is an even function.)
3. The graph of f(x) is symmetric about the origin. (This is false as a function being symmetric about the origin indicates it is odd.)

Therefore, the true statement is:

**The graph of f(x) is symmetric about the y-axis.**