The image presents a graph of a function $y = f(x)$ along with several statements. Here's a detailed analysis:

Graph Description:

• The graph shows a curve starting from the third quadrant, crossing the origin (0,0), and then increasing in the first quadrant.
• The function appears to have a positive slope as $x$ increases after $x = 0$.
• The function is decreasing for values of $x$ less than 0.

Statements to Evaluate:

I. $y = f(x)$ function is increasing.

• This is true for $x > 0$, where the graph has a positive slope.

II. $y = f(x)$ function decreases in the interval $(-\infty, 0]$.

• This is true as the graph is decreasing for values of $x$ less than or equal to 0.

III. $y = f(x)$ function increases for all $x$ values greater than 0.

• This is true, as the graph shows the function increasing for $x > 0$.

Conclusion:

Based on the analysis, the correct statements are:

• I (increasing for $x > 0$),
• II (decreasing for $x < 0$),
• III (increasing for all $x > 0$).

Thus, the correct answer is E) I, II, and III.

Would you like further details or have any questions? Here are five related questions to expand your understanding:

1. How can you identify if a function is increasing or decreasing from its graph?
2. What is the significance of the slope of the tangent to the curve at different points?
3. What is a monotonic function, and how does it apply here?
4. How do local minima and maxima relate to the increasing/decreasing behavior of functions?
5. What are the implications of symmetry in the graph of a function?

Tip: Always check the slope of the function in different intervals to determine whether it is increasing or decreasing.