Question

what is the range of the absolute value function below? (graph of an absolute value function with vertex at (4,1), crossing x - axis at (3,0) and (5,0), y - intercept at (0, - 3)) options: \\( f(x) \geq - 4 \\), \\( f(x) \geq - 1 \\), \\( f(x) \leq 1 \\), \\( f(x) \leq 4 \\)

Explanation:

Step1: Recall Range Definition

The range of a function is the set of all possible output values (y - values) it can take. For a graph, we look at the vertical extent (minimum and maximum y - values).

Step2: Analyze the Graph's Vertex and Direction

The graph is an absolute - value function (a V - shaped graph, but here it's a triangle - like graph which is a transformed absolute - value graph). The vertex of the graph is at (4, 1). The graph opens downward (since the lines go from the vertex downwards on both sides). So, the maximum y - value of the function is 1 (at the vertex), and as x moves away from 4 (in both directions), the y - values decrease without bound? Wait, no, wait. Wait, looking at the graph again: the left - hand side of the graph (for x < 4) is a line with a positive slope, and the right - hand side (for x > 4) is a line with a negative slope. The highest point (the vertex) is at y = 1, and then as x moves to the left or right from x = 4, the y - values go down. Wait, but also, when we look at the lower part, does the graph have a lower bound? Wait, no, the left - hand line goes downwards as x decreases, and the right - hand line also goes downwards as x increases. Wait, but wait, the options are about inequalities. Wait, the vertex is at (4,1), and the graph is decreasing on both sides of x = 4. So the maximum value of f(x) is 1, and then f(x) can be at most 1, and then it gets smaller (but can it go to negative infinity? Wait, but the options are: f(x)≥ - 4, f(x)≥ - 1, f(x)≤1, f(x)≤4. Wait, maybe I misread the graph. Wait, let's check the y - axis. The vertex is at y = 1. The graph comes down from y = 1, and when x is 0, y is - 3? Wait, no, the y - axis: the grid lines. Wait, the vertex is at (4,1). Then, for the range, since the graph has a maximum at y = 1 and then decreases, so all the y - values of the function are less than or equal to 1. Let's check the options:

• Option 1: f(x)≥ - 4. The graph goes below y=-4? Wait, no, the left - hand line: when x = 0, y=-3 (from the graph, the y - intercept is at (0, - 3)). As x decreases, y decreases further. But the option f(x)≤1: since the maximum y - value is 1, and then y can be at most 1, and then less than 1. So the range is all real numbers y such that y≤1, or f(x)≤1.

Answer:

$f(x)\leq1$ (the option corresponding to "f(x) ≤ 1")