Evaluate each of the functions below at x = 1, 2, 4, 8, and 16. Plot the graph of each function. Classify each as linear, quadratic, polynomial, exponential, or logarithmic, and explain the reasons for your classifications. Compare how quickly each function increases, based on the evaluations and graphs, and rank the functions from fastest to slowest growing.
F(x)=ex
E1 = 2.72
E2 = 7.39
E4 = 54.6
E8= 2980.96
E16= 8886111
How would you graph this?
To evaluate the function f(x) = ex at x = 1, 2, 4, 8, and 16, we can use the given values of e:
f(1) = e1 = 2.72
f(2) = e2 = 7.39
f(4) = e4 = 54.6
f(8) = e8 = 2980.96
f(16) = e16 = 8886111
To plot the graph of the function f(x) = ex, we can create a table with values of x and their corresponding f(x) values, and then plot those points on a graph.
x | f(x)
--------------
1 | 2.72
2 | 7.39
4 | 54.6
8 | 2980.96
16 | 8886111
The graph of f(x) = ex will be an exponential curve that starts from the point (0,1) and increases rapidly as x increases.
Classifying the function f(x) = ex, we can say that it is an exponential function. Exponential functions can be recognized by the presence of a constant base raised to a variable power. In this case, the base is e ≈ 2.71828 (the natural logarithm base) and the variable power is x.
Comparing the growth rates of the functions evaluated and their corresponding graphs, we can rank them from fastest to slowest growing:
1. f(x) = ex: The graph of this exponential function increases rapidly as x increases. It grows faster than any linear, quadratic, or polynomial function. This is evident from the evaluations at x = 1, 2, 4, 8, and 16, where the f(x) values increase exponentially.
Therefore, the rank of the functions from fastest to slowest growing would be:
f(x) = ex (exponential) > (any other function)
To evaluate the function f(x) = e^x at x = 1, 2, 4, 8, and 16, you can substitute those values into the function and compute the result. In this case, e represents the mathematical constant known as Euler's number, which is approximately 2.71828.
1. Substitute x = 1 into the function f(x) = e^x:
f(1) = e^1 = e = 2.71828
2. Substitute x = 2 into the function:
f(2) = e^2 ≈ 7.38906
3. Substitute x = 4 into the function:
f(4) = e^4 ≈ 54.59815
4. Substitute x = 8 into the function:
f(8) = e^8 ≈ 2980.95799
5. Substitute x = 16 into the function:
f(16) = e^16 ≈ 8886110.52051
To graph the function f(x) = e^x, you need to create a coordinate system with x and f(x) or y. On the x-axis, you can mark the values 1, 2, 4, 8, and 16. On the y-axis, you need to plot the corresponding function values calculated earlier: 2.72, 7.39, 54.6, 2980.96, and 8886111.
Since the graph of f(x) = e^x is an exponential function, its shape resembles a curve that starts near the origin and increases rapidly as x becomes larger. As x increases, the function value grows at an increasing rate.
Based on the evaluations and the graph, the function f(x) = e^x is classified as an exponential function since it involves the base of Euler's number, e, raised to the power of x. Exponential functions have a fixed base, with x serving as the exponent. In this case, the base is e.
Comparing the growth rates, you can observe that as x increases, the function values increase exponentially:
2.72 < 7.39 < 54.6 < 2980.96 < 8886111
Thus, we can rank the functions from fastest to slowest growing as follows:
1. f(x) = e^x, with E16 = 8886111
2. f(x) = e^x, with E8 ≈ 2980.96
3. f(x) = e^x, with E4 ≈ 54.6
4. f(x) = e^x, with E2 ≈ 7.39
5. f(x) = e^x, with E1 = 2.72