Question

the graph is a piecewise linear function with a vertex at (2, -4) (assuming the grid is 1 unit per square, adjusting for possible y - axis labeling clarity). the options are: ( x leq -4 ), ( x geq 2 ), ( (-infty, -4) ), ( (-infty, +infty) ).

Explanation:

Step1: Analyze the graph's domain

The graph is a piecewise linear function (a V - shaped graph, likely an absolute - value - related graph). The domain of a function is the set of all possible x - values for which the function is defined. Looking at the graph, we can see that the graph extends infinitely to the left and infinitely to the right. There are no breaks or restrictions on the x - values that the graph can take.

Step2: Evaluate each option

• Option 1: \(x\leq - 4\) suggests that the function is only defined for x less than or equal to - 4, but the graph goes beyond x=-4 (both to the left of - 4 and to the right of - 4), so this is incorrect.
• Option 2: \(x\geq2\) suggests the function is only defined for x greater than or equal to 2, but the graph exists for x less than 2, so this is incorrect.
• Option 3: \((-\infty,-4)\) suggests the function is only defined for x in the interval from negative infinity to - 4, but the graph is defined for x outside of this interval (e.g., x = 0, x = 2, etc.), so this is incorrect.
• Option 4: \((-\infty,+\infty)\) means the function is defined for all real numbers. Since the graph extends infinitely in both the positive and negative x - directions, this is correct.

Answer:

\((-\infty,+\infty)\)