Question

identify whether the graph of the function $f(x)$ is even, odd, or neither. this graph is even because it is symmetrical over the $y$-axis. this graph is neither because it is not symmetrical over the $y$-axis or about the origin. this graph is even because it is symmetrical about the origin. this graph is odd because it is symmetrical about the origin.

Explanation:

Step1: Recall definitions of even/odd functions

An even function's graph is symmetric about the \( y \)-axis (i.e., \( f(-x)=f(x) \)). An odd function's graph is symmetric about the origin (i.e., \( f(-x) = -f(x) \)).

Step2: Analyze the graph's symmetry

Looking at the graph, check symmetry over \( y \)-axis: If we reflect the graph over the \( y \)-axis, the left and right sides match. So it's symmetric over \( y \)-axis.

• Check options:
• First option: Says even (symmetric over \( y \)-axis) – matches.
• Second option: Says neither – incorrect, as it is symmetric over \( y \)-axis.
• Third option: Says even due to origin symmetry – origin symmetry is for odd, so incorrect.
• Fourth option: Says odd (origin symmetry) – graph is not symmetric about origin, so incorrect.

Answer:

This graph is even because it is symmetrical over the \( y \)-axis.