Question

if the graph of $f(x)$ is:
which of the following is the graph of $-f(-x)$?
○ a
○ b
○ c

Explanation:

Step1: Analyze transformation \( y = -f(-x) \)

The transformation \( y = -f(-x) \) consists of two parts: reflection over the \( y \)-axis (due to \( -x \)) and reflection over the \( x \)-axis (due to the negative sign in front of \( f \)). First, reflecting \( f(x) \) over the \( y \)-axis gives \( f(-x) \), then reflecting \( f(-x) \) over the \( x \)-axis gives \( -f(-x) \).

Step2: Apply reflections to the original graph

1. Reflection over \( y \)-axis: The original graph of \( f(x) \) has a peak near the origin and then decreases. Reflecting over the \( y \)-axis would flip the graph horizontally, so the part on the right of the \( y \)-axis moves to the left and vice - versa.
2. Reflection over \( x \)-axis: After the horizontal reflection, reflecting over the \( x \)-axis flips the graph vertically, so the positive \( y \)-values become negative and vice - versa.

Looking at the options:

• Option A: Does not show the correct combination of horizontal and vertical reflections.
• Option B: When we first reflect the original \( f(x) \) over the \( y \)-axis (flipping left - right) and then over the \( x \)-axis (flipping up - down), the graph in option B matches this transformation.
• Option C: Does not match the transformation of \( -f(-x) \).

Answer:

B