What is meant when saying a function has a solution?

Question

I have to code the solutions to these questions in Python. But before even thinking about coding, I have to understand what I have to do here.

Table 1: Observed frequency distribution of Y $$\begin{array}{|c|c|c|}\hline \text{Y} & 1 & 2 & 3& 4& 5& 6& 7& 8& 9+\\ \hline \text{Frequency} & 700 & 205& 50&26&10&6&1&1&1 \\ \hline \end{array}$$

$a)$Define $\bar\theta$ as the solution to the following function $$f(\theta)=\bar y +\frac{\theta}{(1-\theta)log(1-\theta)},$$ where $\bar y$ is the mean of the values in Table 1, that is, $f(\bar\theta)=0$.

What is meant when saying a function has a solution?

$b)$ Show that the condition $f(\bar\theta)=0$ is unique. (Hint: draw a graph.)

$c)$ Set up the algorithm you would use to solve for $\bar\theta$. Does anyone have an idea where to start?

Answer 1

Re your first question: This is poor language. Functions do not have solutions, equations do. Functions have zeros. The definition of $\bar\theta$ should be:

Define $\bar\theta$ as the zero of the function $$ f(\theta)=\bar y +\frac{\theta}{(1-\theta)log(1-\theta)}, $$ where $\bar y$ is the mean of the values in Table 1. In other words, $\bar\theta$ is the solution to the equation $f(\bar\theta) = 0$.