Consider the function f: R → R, f(x) = x e(-x).Draw the graph of y = x and y = e–x on the same set of axes, and show how the graph of f could be obtained from these graphs.
Question
Consider the function f: R → R, f(x) = x e(-x).Draw the graph of y = x and y = e–x on the same set of axes, and show how the graph of f could be obtained from these graphs.
Solution
To answer this question, you would need to follow these steps:
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Draw the graph of y = x: This is a straight line that passes through the origin (0,0) and has a slope of 1. It extends infinitely in both the positive and negative directions.
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Draw the graph of y = e^(-x): This is an exponential decay curve. At x = 0, y = 1. As x increases, y approaches 0, and as x decreases, y increases without bound.
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To obtain the graph of f(x) = x * e^(-x), you would need to consider each x-value on the graphs of y = x and y = e^(-x), and multiply the corresponding y-values together. This is because for each x in the domain of f, f(x) is the product of x (from the line y = x) and e^(-x) (from the curve y = e^(-x)).
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The graph of f(x) = x * e^(-x) starts at the origin (0,0). For positive x, it first increases as x starts to dominate the decay of e^(-x), reaches a maximum, and then decreases towards 0 as e^(-x) decays to 0. For negative x, f(x) stays at 0 because the product of a negative number (from y = x) and a positive number (from y = e^(-x)) is negative, but we are considering the real function f: R → R, which means we are only considering real (and in this case, non-negative) values of f(x).
Please note that this is a qualitative description of the graphs. For a precise graph, you would need to use graphing software or a graphing calculator.
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