To find the domain and range of the function shown in the image, we can analyze the graph:

1. Domain:

The domain refers to the set of all possible input values (x-values) for which the function is defined.

• From the graph, the parabola extends indefinitely to the left and right, meaning there are no restrictions on the x-values. Thus, the domain is: $\text{Domain} = (-\infty, \infty)$

2. Range:

The range refers to the set of all possible output values (y-values) that the function can take.

• The lowest point of the parabola, the vertex, appears to be at $y = 0$, and the parabola opens upwards. Therefore, the function takes values from $y = 0$ and increases indefinitely. Thus, the range is: $\text{Range} = [0, \infty)$

Would you like more details on how to derive the domain and range from a graph, or do you have any other questions? Here are some related questions to deepen your understanding:

1. How do we determine if a function's graph extends to infinity?
2. What is the significance of the vertex in a quadratic function?
3. How can we algebraically determine the domain and range of a function without graphing?
4. How does the direction a parabola opens affect its range?
5. What types of functions have limited domains?

Tip: When analyzing a quadratic function from its graph, the vertex is a key feature in determining the range, especially for parabolas.