Question

identify the domain and range for the following function:
graph of a parabola with x-axis from 0 to 6 and y-axis from -5 to 1, with two red arrows at the top ends of the parabola, and two dropdown menus labeled range and domain with choose... options

Explanation:

Step1: Determine the Domain

The domain of a function is the set of all possible x - values. From the graph, we can see that the function starts at \(x = 1\) (the left - most point) and ends at \(x = 5\) (the right - most point). So, the domain is the set of all real numbers \(x\) such that \(1\leq x\leq5\). In interval notation, the domain is \([1,5]\).

Step2: Determine the Range

The range of a function is the set of all possible y - values. The graph of the function is a parabola opening upwards. The minimum value of \(y\) occurs at the vertex of the parabola. From the graph, we can see that the lowest point (the vertex) has a \(y\) - value of \(- 4\), and the highest points (the endpoints) have a \(y\) - value of \(1\). So, the range is the set of all real numbers \(y\) such that \(-4\leq y\leq1\). In interval notation, the range is \([-4,1]\).

Answer:

Domain: \([1,5]\)
Range: \([-4,1]\)