Graph e^x

Problem

ex

Solution

1. Identify the function as the natural exponential function, where the base is e≈2.718

2. Determine the domain and range. The domain is all real numbers, (−∞,∞) and the range is all positive real numbers, (0,∞)

3. Find the y-intercept by evaluating the function at x=0 Since e0=1 the graph passes through (0,1)

4. Identify the horizontal asymptote. As x approaches negative infinity, ex approaches 0 so the line y=0 (the x-axis) is a horizontal asymptote.

5. Observe the growth behavior. As x increases, the function grows rapidly toward infinity. For example, at x=1 y=e≈2.718

6. Sketch the curve starting near the x-axis on the left, passing through (0,1) and (1,e) and curving sharply upward to the right.

Final Answer

The graph of ƒ(x)=ex is a strictly increasing curve with a y-intercept at (0,1) and a horizontal asymptote at y=0

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