To verify that $f(x)$ is a probability mass function (PMF), we need to ensure that:

1. $f(x) \geq 0$ for all $x$.
2. The sum of all $f(x)$ values equals 1.

However, you provided incomplete information regarding $f(x)$. I would need the specific values of $f(x)$ for $x = 0, 1, 2, 3, 4$ to determine if it meets the criteria for a PMF. Once verified, I can calculate the probabilities for the specified events:

Probabilities to Calculate

Assuming $f(x)$ is a valid PMF, I will compute:

1. $P(X = 4) = f(4)$
2. $P(X \leq 2) = f(0) + f(1) + f(2)$
3. $P(2 \leq X < 4) = f(2) + f(3)$
4. $P(X \geq 2) = f(2) + f(3) + f(4)$

Please provide the values of $f(x)$ for each $x$ so I can proceed with the solution.

Would you like details on probability mass functions or have specific questions?

Here are five related questions for further exploration:

1. How does a PMF differ from a probability density function (PDF)?
2. What are the necessary conditions for a function to be considered a PMF?
3. How can we interpret the cumulative distribution function (CDF) in discrete cases?
4. What role does the expectation of a discrete random variable play in probability theory?
5. How do we determine variance for a discrete random variable?

Tip: Remember, the sum of all values in a PMF must equal 1; this helps ensure the function is a true probability distribution.