The graph shown in the image is a function that displays symmetry about the y-axis and several oscillatory features, which suggests that the function depends on the absolute value of $x$ and involves exponential decay or growth. Let's analyze the options based on the visual clues:

Key features from the graph:

1. The function appears to exhibit exponential decay as $x$ increases in both directions from the origin.
2. The graph is symmetric about the y-axis, suggesting the involvement of $|x|$.
3. There are dips and peaks near $x = 0$, indicating oscillatory behavior likely modulated by an exponential term.

Evaluation of each option:

• Option (A): $f(x) = x^2 \cdot 2^{-|x|}$

  • This function includes $x^2$, which is symmetric about the y-axis.
  • The factor $2^{-|x|}$ introduces exponential decay for increasing $|x|$.
  • This matches the symmetry and decaying pattern in the graph, along with the dip near $x = 0$.

• Option (B): $f(x) = x \cdot 2^{-|x|}$

  • This function has the term $x$, which would result in a non-symmetric graph about the y-axis, since $x$ changes sign on different sides of the y-axis.
  • This does not match the observed symmetry in the graph.

• Option (C): $f(x) = |x| \cdot 2^{-|x|}$

  • This function involves $|x|$, maintaining the symmetry about the y-axis.
  • However, the graph does not seem to have a simple linear behavior for small values of $|x|$; instead, there are peaks and dips.
  • This option seems close but does not fully explain the oscillations.

• Option (D): $f(x) = x \cdot 2^{-x}$

  • This function is neither symmetric about the y-axis nor does it account for oscillatory behavior.
  • It can be immediately ruled out based on the lack of symmetry.

Conclusion:

The graph best matches Option (A), $f(x) = x^2 \cdot 2^{-|x|}$, as it explains the symmetry, the exponential decay, and the oscillatory dips near $x = 0$.

Would you like further clarification or details?

Related Questions:

1. How does the behavior of $f(x) = x^2 \cdot 2^{-|x|}$ change as $x$ approaches infinity?
2. How does the absolute value function affect the symmetry of functions like $f(x)$?
3. What is the role of exponential decay in functions involving $2^{-|x|}$?
4. How does squaring the $x$ term influence the shape of the graph near $x = 0$?
5. How do functions with different bases for exponential terms (like 2 or $e$) affect the graph's shape?

Tip:

When analyzing graphs for symmetry, always check if the function involves absolute values or even powers of $x$. These ensure the graph is symmetric about the y-axis.