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:

If a function’s graph is symmetric with respect to the $y$-axis, then $f(x) = f(-x)$ and it is an even function.

Answer 1

The Correct Option is B

From the graph, we observe that $f(x)$ is a horizontal line at $y = 2$ for all $x$. For any $x$, \[ f(x) = 2 \quad \text{and} \quad f(-x) = 2 \] Thus $f(x) = f(-x)$ for all $x$. This is the definition of an even function. Options (1) and (4) would require different scaling between $f(x)$ and $f(-x)$, which is not the case here. Option (3) implies $f$ is odd, which would require $f(x) = -f(-x)$, impossible for a constant nonzero function. Hence, \[ \boxed{f(x) = f(-x)} \]