Find the associated cumulative distribution function. Graph both functions (on separate sets of axes). \[ f(x)=\left\{\begin{array}{ll} \frac{1}{8}+\frac{1}{512} x^{3} & \text { if }-4 \leq x \leq 4 \\ 0 & \text { otherwise } \end{array}\right. \]

Step 1 ：The problem provides a probability density function (PDF) \(f(x)\) defined as follows: \[f(x)=\left\{\begin{array}{ll} \frac{1}{8}+\frac{1}{512} x^{3} & \text { if } -4 \leq x \leq 4 \\ 0 & \text { otherwise } \end{array}\right.\]

Step 2 ：The cumulative distribution function (CDF) of a random variable is defined as the probability that the variable takes a value less than or equal to a certain value. The CDF is the integral of the PDF.

Step 3 ：We need to find the integral of \(f(x)\) from -infinity to x to get the CDF. The integral of \(f(x)\) is \(F(x) = 0.00048828125*x^{4} + 0.125*x\).

Step 4 ：However, the CDF should be defined for all real numbers. The function \(f(x)\) is only defined for x in the interval [-4, 4]. For x < -4 or x > 4, the CDF should be 0 or 1, respectively. Therefore, we need to define the CDF as a piecewise function.

Step 5 ：The cumulative distribution function (CDF) is given by: \[F(x)=\left\{\begin{array}{ll} 0 & \text { if } x < -4 \\ \frac{1}{2048} x^{4}+\frac{1}{8} x & \text { if } -4 \leq x \leq 4 \\ 1 & \text { if } x > 4 \end{array}\right.\]

Step 6 ：\(\boxed{F(x)=\left\{\begin{array}{ll} 0 & \text { if } x < -4 \\ \frac{1}{2048} x^{4}+\frac{1}{8} x & \text { if } -4 \leq x \leq 4 \\ 1 & \text { if } x > 4 \end{array}\right.}\) is the final answer.