The table below represents the function f(x)

Question

the two catogories are x, and f(x)
x, -5, -3, -1, 0, 1, 3,
f(x), 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.
The graph of f of x is symmetric about the origin.

Answer 1

To determine the symmetry of the function \( f(x) \), we need to consider the definition of an even function. A function \( f(x) \) is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \).

Given the values from the table:

• \( f(-5) = 7 \) and \( f(5) \) is not provided, but we won't use this point for testing symmetry.
• \( f(-3) = 3 \) and \( f(3) = 3 \) → satisfies \( f(-3) = f(3) \)
• \( f(-1) = -2 \) and \( f(1) = -2 \) → satisfies \( f(-1) = f(1) \)
• \( f(0) = 1 \) (symmetric about the y-axis but only at x = 0)

Since \( f(-x) = f(x) \) holds true for the values provided (except for \( f(5) \)), we can conclude that the function \( f(x) \) indeed demonstrates even symmetry about the y-axis.

Thus, the correct statement is:

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