which function has a range limited to only negative numbers?

Explanation:

Step1: Recall Range Definition

The range of a function is the set of all possible ( y )-values (outputs) the function can take. We need to analyze the given graph (and potentially other functions, though only this graph is shown here) to see which has a range with only negative numbers (i.e., ( y < 0 ) for all points on the graph).

Step2: Analyze the Given Graph

Looking at the orange graph:

• The vertex (highest point) of the graph is at ( (2, -2) )? Wait, no, wait. Wait, the grid: let's check the coordinates. Wait, the ( y )-axis: the top of the graph (the peak) is at ( x = 2 ), and what's the ( y )-value? Wait, the grid lines: the ( y )-axis has marks at 2, -2, -4, -6, -8. Wait, the peak of the graph (the highest point) is at ( (2, -2) )? Wait, no, looking at the graph, the line goes up to ( (2, -2) )? Wait, no, maybe I misread. Wait, the ( y )-axis: the positive ( y ) is up, negative down. The graph's highest point (the vertex) is at ( (2, -2) )? Wait, no, actually, let's check the coordinates. Wait, the graph starts from the bottom left (going up) to ( (2, -2) ) (the peak) then down. Wait, but ( -2 ) is negative. Wait, all the ( y )-values on the graph: let's see, the lowest points are more negative, but the highest point (the peak) is at ( y = -2 ), which is negative. So all points on this graph have ( y )-values less than or equal to ( -2 ) (wait, no, wait, maybe I messed up the grid. Wait, maybe the peak is at ( (2, -2) ), but actually, looking at the graph, the line crosses the ( y )-axis at ( (0, -3) ) (since between -2 and -4, maybe -3). Wait, no, the key is: does any point on the graph have a ( y )-value ( \geq 0 )? Looking at the graph, the highest point (the vertex) is at ( (2, -2) ), which is negative. So all ( y )-values on this graph are negative (since the peak is at ( y = -2 ), and it goes down from there). So the range of this function is all real numbers ( y ) such that ( y \leq -2 ), which are all negative numbers (since ( -2 < 0 ), and all lower ( y )-values are more negative).

Assuming this is the function in question (since it's the only graph shown), this graph's range is limited to negative numbers (because the highest ( y )-value it reaches is ( -2 ), which is negative, so all ( y )-values on the graph are negative).

Answer:

The function represented by the given orange graph (the piece - wise linear graph with vertex at ( (2, - 2) ) and extending to more negative ( y )-values) has a range limited to only negative numbers.