Question

identify whether the graph of the function $f(x)$ shown below is even, odd, or neither.

the graph is because

Explanation:

Step1: Recall definitions of even/odd functions

An even function satisfies \( f(-x) = f(x) \) (symmetric about \( y \)-axis). An odd function satisfies \( f(-x) = -f(x) \) (symmetric about origin).

Step2: Analyze the graph's symmetry

Visually check the graph. For a point \( (x, y) \) on the graph, check \( (-x, y) \) (for even) or \( (-x, -y) \) (for odd). The graph is symmetric about the \( y \)-axis? No. Wait, wait—wait, looking at the graph: let's see, when \( x \) is positive and negative. Wait, actually, let's check symmetry about \( y \)-axis: if we reflect over \( y \)-axis, does it match? Wait, no—wait, maybe I made a mistake. Wait, no, let's re-express: the graph of an even function is symmetric with respect to the \( y \)-axis. The graph of an odd function is symmetric with respect to the origin. Let's check points. For example, take \( x = 3 \): what's \( f(3) \)? It's 0? Wait, no, the graph: let's see, the right side (positive \( x \)) and left side (negative \( x \)). Wait, when \( x \) is positive, say \( x = 2 \), \( f(2) \) is some value, and \( x = -2 \): does \( f(-2) = f(2) \)? Wait, no, wait the graph: looking at the left and right. Wait, actually, the graph is symmetric about the \( y \)-axis? Wait, no, wait the left side: when \( x \) is negative, the line goes down, and right side when \( x \) is positive, the line goes down. Wait, no, maybe it's symmetric about the \( y \)-axis? Wait, no, let's think again. Wait, the definition: even function: symmetric over \( y \)-axis. Odd: symmetric over origin. Let's check the graph. If we fold the graph over the \( y \)-axis, does the left half match the right half? Let's see, the peak at \( x = 0 \) area, and the lines on left and right. Wait, actually, the graph is symmetric about the \( y \)-axis? Wait, no, wait the left side: when \( x \) is -3, what's \( f(-3) \)? Let's see, the left line: when \( x = -3 \), maybe \( f(-3) = 0 \)? And \( x = 3 \), \( f(3) = 0 \). Then, for \( x = 2 \), \( f(2) \) is the same as \( f(-2) \)? Wait, maybe the graph is even? Wait, no, wait the slope: left side, when \( x \) increases (towards 0), the function increases, and right side, when \( x \) increases (away from 0), the function decreases. Wait, maybe I messed up. Wait, no—wait, the key is: even function: \( f(-x) = f(x) \). So if the graph is symmetric over \( y \)-axis, it's even. Let's check the graph. The given graph: the right half (positive \( x \)) and left half (negative \( x \))—do they mirror over \( y \)-axis? Yes, because the shape on the left (for negative \( x \)) is the mirror image of the right (positive \( x \)) over the \( y \)-axis. So that means \( f(-x) = f(x) \), so it's even.

Answer:

The graph is even because it is symmetric about the \( y \)-axis (satisfies \( f(-x) = f(x) \)).