Evaluate the exponential function for three positive values of x, three negative values of x, and at x=0. Show your work. Use the resulting ordered pairs to plot the graph; submit the graph via the Dropbox. State the domain and the range of the function.
f(x) = e^(-x) - 1
To evaluate the exponential function f(x) = e^(-x) - 1 for various values of x, we substitute the given values of x into the function and calculate the corresponding values of f(x).
Let's evaluate f(x) for three positive values of x:
1. For x = 1: f(1) = e^(-1) - 1 = 0.3679 - 1 = -0.6321
2. For x = 2: f(2) = e^(-2) - 1 = 0.1353 - 1 = -0.8647
3. For x = 3: f(3) = e^(-3) - 1 = 0.0498 - 1 = -0.9502
Now, let's evaluate f(x) for three negative values of x:
1. For x = -1: f(-1) = e^-(-1) - 1 = e^(1) - 1 = 2.7183 - 1 = 1.7183
2. For x = -2: f(-2) = e^-(-2) - 1 = e^(2) - 1 = 7.3891 - 1 = 6.3891
3. For x = -3: f(-3) = e^-(-3) - 1 = e^(3) - 1 = 20.0855 - 1 = 19.0855
Finally, let's evaluate f(x) at x = 0: f(0) = e^(0) - 1 = 1 - 1 = 0
The resulting ordered pairs, with x as the independent variable and f(x) as the dependent variable, are as follows:
Positive values of x: (1, -0.6321), (2, -0.8647), (3, -0.9502)
Negative values of x: (-1, 1.7183), (-2, 6.3891), (-3, 19.0855)
x = 0: (0, 0)
To plot the graph, plot the above-mentioned ordered pairs on a coordinate plane. The x-values represent the horizontal axis, and f(x) values correspond to the vertical axis. Connect the points to obtain a smooth curve. Note that the graph should approach but never touch the x-axis.
The domain of the function f(x) = e^(-x) - 1 is all real numbers since the exponential function is defined for every value of x.
The range of the function f(x) = e^(-x) - 1 is (-∞, -1), which means all values less than -1.