Question

Graphs of functions are given. Mark option

• A. If f(x) = 3f(−x)
• B. If f(x) = f(−x)
• C. If f(x) = −f(−x)
• D. If 3f(x) = 6f(−x)

Hint:

When comparing $f(x)$ and $f(-x)$ from graphs, pick symmetric points and compute their values to detect proportionality.

Answer 1

The Correct Option is A

From the graph we observe:

• For \( x > 0 \), the function increases linearly with slope \( +1 \) starting from the origin. Therefore: \[ f(1) = 1 \]
• For \( x < 0 \), the function decreases with slope \( -2 \), also starting from the origin. Therefore: \[ f(-1) = 2 \]

Step 1: Compare Values at \( x = 1 \) and \( x = -1 \)

Using the values: \[ f(1) = 1, \quad f(-1) = 2 \] We find a relationship: \[ f(-1) = 2f(1) \quad \Rightarrow \quad f(1) = \frac{1}{2} f(-1) \]

Step 2: Evaluate the Given Options

Suppose the function satisfies \( f(x) = 3f(-x) \). Then: \[ f(1) = 3f(-1) \quad \Rightarrow \quad 1 = 3 \times 2 = 6 \quad \text{(Incorrect)} \] Now try: \[ f(x) = \frac{1}{2} f(-x) \quad \Rightarrow \quad f(1) = \frac{1}{2} \times f(-1) = \frac{1}{2} \times 2 = 1 \quad \text{(Correct)} \] So this relation works. That means: \[ f(-x) = 2f(x) \] Which is equivalent to: \[ \boxed{f(x) = \frac{1}{2}f(-x)} \] So the best matching option among the choices is the one where: \[ f(x) = \frac{1}{2} f(-x) \] or equivalently: \[ f(-x) = 2 f(x) \]

Conclusion

Therefore, the correct functional relationship supported by the graph is: \[ \boxed{f(x) = \frac{1}{2} f(-x)} \quad \text{or} \quad \boxed{f(-x) = 2f(x)} \] Among multiple-choice options, this matches the option where the **negative side is scaled by 2** relative to the positive.