Here are the plots for the function $f(z) = 2z^2 \left(1 - z^2\right)^{-\frac{1}{2}} - 1$:

1. Plot for $0 < z < 10^{-7}$ (left plot): The function is shown over this small interval, filling the horizontal axis.
2. Plot for $0 < z < 10^{-15}$ (right plot): This shows the function over an even smaller interval, as requested.

Let me know if you'd like a detailed explanation of the behavior of the function in these intervals or if you have other questions.

Here are five related questions to consider:

1. What happens to the function $f(z)$ as $z$ approaches zero?
2. How does the behavior of $f(z)$ change if we change the power in the denominator $\left(1 - z^2\right)^{-1/2}$?
3. What would the function look like for negative values of $z$?
4. How does this function compare to simpler functions like $z^2$?
5. How does the term $\left(1 - z^2\right)^{-\frac{1}{2}}$ affect the overall behavior of $f(z)$?

Tip: Be cautious with very small intervals like $10^{-15}$, as the function can be numerically unstable near zero!