You provided the function:

$f(x) = \frac{1}{3} \times \left(1 - \frac{|x|}{3}\right) \quad \text{for} \quad -3 \leq x \leq 3$

and you're asking for $H(x)$.

Typically, $H(x)$ in such contexts could be the cumulative distribution function (CDF) if this is a probability density function (PDF). To determine $H(x)$, we would integrate the given function $f(x)$ from the lower limit to some value $x$.

Step-by-step outline:

1. The function $f(x)$ is symmetric and defined piecewise for $x \in [-3, 3]$.
2. We need to calculate the integral of $f(x)$ to obtain $H(x)$.

To confirm if you are referring to the cumulative function or if there's another interpretation of $H(x)$, could you clarify this?

Let me know if you'd like me to proceed with integration or provide more details on related functions. Here are 5 potential questions that could follow:

1. Would you like to compute the integral of $f(x)$ to find $H(x)$?
2. Is $H(x)$ supposed to be a step function or a cumulative distribution function?
3. Should we also verify if the function $f(x)$ is normalized (if this is a PDF)?
4. Do you need the plot of $f(x)$ to visualize the behavior of the function?
5. Should we also explore the properties of this function, like its mean or variance?

Tip:

When dealing with absolute values in functions, especially for piecewise integration, remember that the function may change behavior at the points where the absolute value affects the sign, such as $x = 0$ in this case.