Question

determine the domain and range of the function using the graph below.
the domain is:
\\(\circ x \leq 2\\) \\(\circ x \geq 2\\)
\\(\circ x \leq 0\\) \\(\circ\\) all real numbers
the range is:
\\(\circ y \geq 2\\) \\(\circ y \leq 2\\)
\\(\circ\\) all real numbers \\(\circ y \leq 0\\)
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Explanation:

Step1: Determine Domain

The graph is a parabola opening upwards. The left end is at \( x = -3 \) (closed dot) and the right end is at \( x = 2 \) (closed dot)? Wait, no, looking at the graph, the left vertex? Wait, no, the graph's leftmost point is at \( x=-3 \) (the left end of the parabola) and rightmost at \( x = 2 \)? Wait, no, the graph is a parabola with left endpoint at \( x=-3 \) (closed) and right endpoint at \( x = 2 \) (closed). Wait, no, the domain is the set of all x-values the graph covers. Wait, the graph is a parabola (U - shaped) with left end at \( x=-3 \) (closed) and right end at \( x = 2 \)? Wait, no, looking at the grid, the leftmost x is -3 (the left end of the red graph) and rightmost x is 2 (the right end of the red graph). Wait, no, the graph is a parabola that starts at \( x=-3 \) (top left) and ends at \( x = 2 \) (top right), curving down to a minimum at \( x=-1 \) (approx) with y=2. Wait, no, the domain: the x-values from -3 to 2? Wait, no, the options are \( x \leq 2 \), \( x \geq 2 \), \( x \leq 0 \), or all real numbers. Wait, maybe I misread. Wait, the graph: the left end is at \( x=-3 \) (closed) and the right end is at \( x = 2 \) (closed), but the parabola is between x=-3 and x=2? Wait, no, the options include "All real numbers" but the graph is a segment? Wait, no, the graph is a parabola (U - shaped) with left endpoint at \( x=-3 \) (y=10) and right endpoint at \( x=2 \) (y=10), and the minimum at \( x=-1 \) (y=2). So the domain is all x from -3 to 2? But the options don't have that. Wait, maybe the graph is a function where the domain is \( x \leq 2 \)? Wait, no, the left end is at x=-3, so x goes from -3 to 2? Wait, the options are: \( x \leq 2 \), \( x \geq 2 \), \( x \leq 0 \), or all real numbers. Wait, maybe the graph is a parabola that is defined for all x? No, the graph has endpoints. Wait, maybe the graph is a function where the domain is \( x \leq 2 \) (since the right end is at x=2, and to the left of x=2, including x < 2). Wait, the left end is at x=-3, which is less than 2, so the domain is \( x \leq 2 \)? Wait, the options for domain: first option \( x \leq 2 \), second \( x \geq 2 \), third \( x \leq 0 \), fourth all real numbers.

Step2: Determine Range

The range is the set of y-values. The minimum y-value is 2 (at the vertex), and the maximum y-values are at the endpoints (y=10). So the range is \( y \geq 2 \), since the parabola opens upwards (wait, no, the graph is U - shaped, opening upwards? Wait, the vertex is the minimum point. So the range is all y from 2 to 10, so \( y \geq 2 \).

Wait, let's re - evaluate:

Domain: The graph's x - values. The leftmost x is -3, rightmost is 2. But the options include "All real numbers" which is not correct. Wait, maybe the graph is a function where the domain is \( x \leq 2 \) (since the right end is at x=2, and x can be less than or equal to 2). So domain: \( x \leq 2 \).

Range: The y - values. The minimum y is 2 (at the vertex), and y goes up to 10 (at the endpoints). So the range is \( y \geq 2 \).

Answer:

The domain is: \( x \leq 2 \) (from the options, as the graph's rightmost x is 2 and x can be less than or equal to 2)
The range is: \( y \geq 2 \)

So for domain: the correct option is \( \boldsymbol{x \leq 2} \)
For range: the correct option is \( \boldsymbol{y \geq 2} \)