Let's analyze the information provided in the image and determine the answers to each part.

1. Equation of the function:

   • The graph shows a linear function, which typically takes the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
   • Observing the graph, the line passes through the origin $(0, 0)$ and another point, which appears to be $(1, 1)$.
   • Using these two points, we can calculate the slope $m = \frac{\Delta y}{\Delta x} = \frac{1 - 0}{1 - 0} = 1$.
   • Since the y-intercept $b = 0$, the equation of the function is $f(x) = 1 \cdot x + 0$, or simply $f(x) = x$.

2. Domain of the function:

   • For linear functions that extend infinitely in both directions, the domain is all real numbers.
   • Therefore, the domain is $(-\infty, \infty)$.

3. Range of the function:

   • Similarly, because the line extends infinitely in both the positive and negative directions on the y-axis, the range is also all real numbers.
   • Thus, the range is $(-\infty, \infty)$.

Summary:

• Equation: $f(x) = x$
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, \infty)$

Would you like a more detailed explanation of any part? Here are some follow-up questions to consider:

1. How can you determine the slope of a line from a graph?
2. What are the differences between domain and range in mathematical functions?
3. How do you recognize linear functions from graphs?
4. What does it mean if a function has an infinite range?
5. Can linear functions have restricted domains?

Tip: In linear equations, the slope $m$ indicates the steepness and direction of the line, while the intercept $b$ represents the point where the line crosses the y-axis.