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) = x3 - 3x2 - 2x + 1
f(x) = ex
f(x) = 3x - 2
f(x) = log x
f(x) = x2 - 5x + 6
To evaluate each function at the given x-values (1, 2, 4, 8, and 16), we simply substitute these values into the corresponding functions and calculate the resulting outputs. Let's calculate the values for each function:
f(x) = x^3 - 3x^2 - 2x + 1:
For x = 1: f(1) = 1^3 - 3(1)^2 - 2(1) + 1 = -3
For x = 2: f(2) = 2^3 - 3(2)^2 - 2(2) + 1 = -7
For x = 4: f(4) = 4^3 - 3(4)^2 - 2(4) + 1 = 9
For x = 8: f(8) = 8^3 - 3(8)^2 - 2(8) + 1 = 353
For x = 16: f(16) = 16^3 - 3(16)^2 - 2(16) + 1 = 1921
f(x) = e^x:
For x = 1: f(1) = e^1 = e
For x = 2: f(2) = e^2
For x = 4: f(4) = e^4
For x = 8: f(8) = e^8
For x = 16: f(16) = e^16
f(x) = 3x - 2:
For x = 1: f(1) = 3(1) - 2 = 1
For x = 2: f(2) = 3(2) - 2 = 4
For x = 4: f(4) = 3(4) - 2 = 10
For x = 8: f(8) = 3(8) - 2 = 22
For x = 16: f(16) = 3(16) - 2 = 46
f(x) = log(x):
For x = 1: f(1) = log(1) = 0 (logarithm base does not matter, as log(1) is always 0)
For x = 2: f(2) = log(2)
For x = 4: f(4) = log(4)
For x = 8: f(8) = log(8)
For x = 16: f(16) = log(16)
f(x) = x^2 - 5x + 6:
For x = 1: f(1) = 1^2 - 5(1) + 6 = 2
For x = 2: f(2) = 2^2 - 5(2) + 6 = 0
For x = 4: f(4) = 4^2 - 5(4) + 6 = 6
For x = 8: f(8) = 8^2 - 5(8) + 6 = 46
For x = 16: f(16) = 16^2 - 5(16) + 6 = 214
Now, let's classify each of these functions based on their forms:
f(x) = x^3 - 3x^2 - 2x + 1: This is a polynomial function of degree 3, as it contains the variable x raised to the power of 3. It is neither linear nor quadratic.
f(x) = e^x: This is an exponential function, as it contains the base e raised to the power of x.
f(x) = 3x - 2: This is a linear function since it has a constant slope of 3 and no exponents on x.
f(x) = log(x): This is a logarithmic function, as it involves a logarithm with x as the argument.
f(x) = x^2 - 5x + 6: This is a polynomial function of degree 2, and specifically, it is a quadratic function, as it contains the variable x raised to the power of 2.
To compare how quickly each function increases, we can consider their growth rates and observe their graphs. Let's rank them from fastest to slowest growing:
1. Exponential function (f(x) = e^x): Exponential functions grow the fastest, as the value of e^x increases rapidly for positive values of x.
2. Polynomial function (f(x) = x^3 - 3x^2 - 2x + 1): Polynomial functions have a slower growth rate than exponentials but faster than other types. The cubic term dominates this function, leading to both positive and negative values.
3. Quadratic function (f(x) = x^2 - 5x + 6): Quadratic functions increase more slowly than cubic functions but still have a steady growth rate. The parabolic shape of the graph keeps the growth rate relatively consistent.
4. Linear function (f(x) = 3x - 2): Linear functions have a constant growth rate, represented by the slope. In this case, the growth rate is 3, so it increases more slowly than the previous functions mentioned.
5. Logarithmic function (f(x) = log(x)): Logarithmic functions increase the slowest among the given functions. As the input x increases, the values of the logarithmic function increase but at a decreasing rate.