Question

if $f(x)$ is an even function, which statement about the graph of $f(x)$ must be true? it has line symmetry about the line $y = x$. it has line symmetry about the $x$-axis. it has line symmetry about the $y$-axis. it has rotational symmetry about the origin.

Explanation:

An even function is defined as \( f(-x) = f(x) \) for all \( x \) in the domain. Geometrically, this means that for every point \((x, y)\) on the graph, the point \((-x, y)\) is also on the graph. This is the definition of symmetry about the \( y \)-axis.

• Symmetry about \( y = x \) is for inverse functions (not even functions).
• Symmetry about the \( x \)-axis would mean \( f(x)= -f(x) \) (implying \( f(x) = 0 \) for all \( x \), which is not true for all even functions).
• Rotational symmetry about the origin is a property of odd functions (\( f(-x)= -f(x) \)), not even functions.

Answer:

It has line symmetry about the \( y \)-axis.