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; State the domain and the range of the function.
f(x) = e^x
F(0) = e^0 = 1
F(1) = e^1 = 2,718
F(2) = e^2 = 7.39
(0 , 1) , (1 , 2.72) , (2 , 7.4).
Domain: The set of all real numbers.
Range: (0 , +infinity).
To evaluate the exponential function f(x) = e^x for different values of x, we can substitute those values into the function and calculate the result.
Let's start by evaluating the function for three positive values of x: 1, 2, and 3.
1. f(1) = e^1
Using the value of x = 1, we substitute it into the function:
f(1) = e^1 = e ≈ 2.718
2. f(2) = e^2
Using the value of x = 2:
f(2) = e^2 = e * e = e^2 ≈ 7.389
3. f(3) = e^3
Using the value of x = 3:
f(3) = e^3 = e * e * e = e^3 ≈ 20.086
Now, let's evaluate the function for three negative values of x: -1, -2, and -3.
1. f(-1) = e^(-1)
Using the value of x = -1:
f(-1) = e^(-1) ≈ 0.368
2. f(-2) = e^(-2)
Using the value of x = -2:
f(-2) = e^(-2) ≈ 0.135
3. f(-3) = e^(-3)
Using the value of x = -3:
f(-3) = e^(-3) ≈ 0.050
Finally, let's evaluate the function at x = 0:
f(0) = e^0 = 1
Now, we can plot the resulting ordered pairs on a graph:
Positive values of x: (1, 2.718), (2, 7.389), (3, 20.086)
Negative values of x: (-1, 0.368), (-2, 0.135), (-3, 0.050)
x = 0: (0, 1)
The domain of f(x) = e^x is all real numbers (-∞, ∞) because we can substitute any real number into x.
The range of f(x) = e^x is all positive real numbers (0, ∞) because the exponentiation of any real number will result in a positive number.