Answer: The image shows a computer screen displaying a mathematical question that involves comparing the end behavior of two functions, labeled f and g. The functions are depicted on a graph.
The graph illustrates function f as a curve that starts in the top-left quadrant and decreases as it moves towards the right, approaching the x-axis asymptotically. Function g is described in text as "a decreasing exponential function with a y-intercept of 5 and no x-intercept."
The question asks which statement correctly compares the functions f and g, with four options provided:
A. They have different end behavior as x approaches negative infinity (-∞) but the same end behavior as x approaches infinity (∞).
B. They have the same end behavior as x approaches negative infinity (-∞) but different end behavior as x approaches infinity (∞).
C. They have the same end behavior as x approaches negative infinity (-∞) and the same end behavior as x approaches infinity (∞).
D. They have different end behavior as x approaches negative infinity (-∞) and different end behavior as x approaches infinity (∞).
Based on the graph, function f approaches the x-axis as x approaches infinity, which indicates it approaches a value of 0. This is typical of decreasing exponential functions, which function g is also described to be. Therefore, both functions f and g will approach 0 as x approaches infinity (∞), so they have the same end behavior in that direction.
As x approaches negative infinity (-∞), the behavior of function f cannot be clearly determined from the graph, as it is cut off. However, a decreasing exponential function like g typically approaches a horizontal asymptote that is not the x-axis for negative values of x. Since we do not have information on f for negative infinity and g does not have an x-intercept, we cannot determine if they have the same behavior for x approaching negative infinity solely based on the information provided.
Therefore, without the complete graph for function f, we cannot definitively choose an answer. However, if we assume function f is also a typical decreasing exponential function (as it appears on the graph), option C would be correct, as both functions would asymptotically approach a horizontal line (not necessarily the x-axis) as x approaches negative infinity and both would approach zero as x approaches infinity.
If function f is not a typical decreasing exponential function, then option B or D might be correct, depending on f's behavior as x approaches negative infinity, which is not shown. However, the information provided is not sufficient to make a conclusive decision for options B or D.
