Please help, very lost
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 three positive values of x, we will substitute those values into the function and calculate the corresponding y-values. Similarly, we will do the same for three negative values of x and x = 0.
Positive values of x:
1. Let's take x = 1:
f(1) = e^(-1) - 1
≈ 0.368 - 1
≈ -0.632
So, the ordered pair is (1, -0.632).
2. Now, let's take x = 2:
f(2) = e^(-2) - 1
≈ 0.135 - 1
≈ -0.865
The ordered pair is (2, -0.865).
3. For x = 3:
f(3) = e^(-3) - 1
≈ 0.0498 - 1
≈ -0.9502
The ordered pair is (3, -0.9502).
Negative values of x:
1. Taking x = -1:
f(-1) = e^(-(-1)) - 1
≈ e^(1) - 1
≈ 2.718 - 1
≈ 1.718
So, the ordered pair is (-1, 1.718).
2. For x = -2:
f(-2) = e^(-(-2)) - 1
≈ e^(2) - 1
≈ 7.389 - 1
≈ 6.389
The ordered pair is (-2, 6.389).
3. Taking x = -3:
f(-3) = e^(-(-3)) - 1
≈ e^(3) - 1
≈ 20.086 - 1
≈ 19.086
The ordered pair is (-3, 19.086).
At x = 0:
f(0) = e^(-0) - 1
= e^(0) - 1
= 1 - 1
= 0
The ordered pair is (0, 0).
Now, let's plot the graph using these ordered pairs:
(x, y) pairs:
(1, -0.632)
(2, -0.865)
(3, -0.9502)
(-1, 1.718)
(-2, 6.389)
(-3, 19.086)
(0, 0)
The graph will provide a visual representation of the function.
The domain of the function f(x) = e^(-x) - 1 is all real numbers, (-∞, ∞).
The range of the function is (-1, ∞), meaning the y-values are all values greater than or equal to -1.
To evaluate the exponential function f(x) = e^(-x) - 1 for different values of x and to plot its graph, follow these steps:
Step 1: Find the values of f(x) for three positive values of x.
- Choose three positive values of x, such as x = 1, 2, and 3.
- Substitute these values into the function: f(1) = e^(-1) - 1, f(2) = e^(-2) - 1, f(3) = e^(-3) - 1.
- Use a calculator to find the numerical values of e^(-1), e^(-2), and e^(-3), then subtract 1 to calculate the function values.
Step 2: Find the values of f(x) for three negative values of x.
- Choose three negative values of x, such as x = -1, -2, and -3.
- Substitute these values into the function: f(-1) = e^(1) - 1, f(-2) = e^(2) - 1, f(-3) = e^(3) - 1.
- Use a calculator to find the numerical values of e^(1), e^(2), and e^(3), then subtract 1 to calculate the function values.
Step 3: Find the value of f(x) at x = 0.
- Substitute x = 0 into the function: f(0) = e^(0) - 1.
- Use a calculator to find the numerical value of e^(0), which is equal to 1, then subtract 1 to calculate the function value.
Step 4: Plot the graph using the resulting ordered pairs.
- Use a graphing tool or software to plot the points (x, f(x)) obtained from the previous steps.
- Connect the points and draw a smooth curve, if applicable.
Step 5: State the domain and range of the function.
- The domain represents all possible values of x for which the function is defined. In this case, the domain is all real numbers since the exponential function is defined for any value of x.
- The range represents all possible values of f(x) as x varies. In this case, the range is (-∞, -1) since the function is always decreasing and approaches -1 as x approaches infinity.
Remember to submit the graph via the Dropbox as requested.