The question shown in the image is about determining whether a statement about even functions is true or false:

Statement: "Even functions have graphs that are symmetric with respect to the x-axis."

Let’s break this down:

• Definition of Even Function: A function $f(x)$ is called even if for all $x$ in the domain of $f$, $f(-x) = f(x)$. This property implies that the graph of an even function is symmetric with respect to the y-axis, not the x-axis.

So, the given statement is false because the graph of an even function is symmetric with respect to the y-axis.

Now, let’s go through the answer choices:

• A. False because even functions have graphs symmetric with respect to the y-axis.
  This is the correct answer because the correct symmetry is with the y-axis.

• B. False because even functions have graphs that are symmetric with respect to the origin.
  This is incorrect since functions that are symmetric with respect to the origin are odd functions, not even functions.

• C. True if and only if its graph is symmetric with respect to the x-axis.
  This is incorrect because even functions are not symmetric with respect to the x-axis.

• D. False because even functions are symmetric with respect to the line $y = x$.
  This is incorrect because symmetry with respect to the line $y = x$ is not a characteristic of even functions.

Thus, the correct answer is A.

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:

1. What are the differences between even and odd functions?
2. How do you determine if a function is even or odd?
3. What is the graphical significance of odd functions?
4. Can a function be neither even nor odd? Provide an example.
5. How is the symmetry of a function used in graphing?

Tip: To check if a function is even, substitute $-x$ for $x$ in the equation and see if it remains unchanged.