Problem 39 - Precalculus Mathematics for Calculus

Question

Compare the functions \(f(x)=x^{3}\) and \(g(x)=3^{x}\) by evaluating both of them for \(x=0,1,2,3,4,5,6,7,8,9,10,15,\) and 20 Then draw the graphs of \(f\) and \(g\) on the same set of axes.

Step-by-Step Solution

Answer

For small x values, \(f(x)\) is initially greater; at \(x=6\), they are equal and for larger \(x\), \(g(x)\) becomes much larger.

Step 1: Evaluate f(x) for each x value

For the function \(f(x) = x^3\), we compute the cube of each x value. \[\begin{align*}f(0) &= 0^3 = 0, \f(1) &= 1^3 = 1, \f(2) &= 2^3 = 8, \f(3) &= 3^3 = 27, \f(4) &= 4^3 = 64, \f(5) &= 5^3 = 125, \f(6) &= 6^3 = 216, \f(7) &= 7^3 = 343, \f(8) &= 8^3 = 512, \f(9) &= 9^3 = 729, \f(10) &= 10^3 = 1000, \f(15) &= 15^3 = 3375, \f(20) &= 20^3 = 8000.\end{align*}\]

Step 2: Evaluate g(x) for each x value

For the function \(g(x) = 3^x\), we compute \(3\) raised to each x value.\[\begin{align*}g(0) &= 3^0 = 1, \g(1) &= 3^1 = 3, \g(2) &= 3^2 = 9, \g(3) &= 3^3 = 27, \g(4) &= 3^4 = 81, \g(5) &= 3^5 = 243, \g(6) &= 3^6 = 729, \g(7) &= 3^7 = 2187, \g(8) &= 3^8 = 6561, \g(9) &= 3^9 = 19683, \g(10) &= 3^{10} = 59049, \g(15) &= 3^{15} = 14348907, \g(20) &= 3^{20} = 3486784401.\end{align*}\]

Step 3: Compare results for f(x) and g(x)

Compare the computed values of both functions:- At \(x = 0\), \(f(0) = 0\) and \(g(0) = 1\)- From \(x = 1\) to \(x = 5\), \(f(x)\) values are smaller than \(g(x)\) values.- At \(x = 6\), both functions give the same value of 729.- From \(x = 7\) onwards, \(g(x)\) greatly surpasses \(f(x)\).

Step 4: Graph both functions

To graph the functions, plot the x values on the x-axis and their corresponding \(f(x)\) and \(g(x)\) values on the y-axis. You'll see that initially \(f(x)\) starts with lower values, matches \(g(x)\) at \(x=3\), and remains lower than \(g(x)\) thereafter. These graphs illustrate the rapid growth of exponential functions compared to polynomial functions as x increases.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of variables with non-negative integer exponents. The most straightforward form of a polynomial is one with a single term, also known as a monomial. As polynomials increase in complexity, they can include multiple terms.

• For example, the function \( f(x) = x^3 \) is a polynomial function of degree 3, which is also referred to as a cubic function.
• Degree signifies the highest power of the variable in the polynomial, which in this case, is 3.

Polynomial functions are known for their smooth and continuous curves when plotted on a graph. They can change direction based on their degree and leading coefficients.
Higher-degree polynomials can have more twists and turns.
As seen in \( f(x) = x^3 \), small values of \(x\) yield relatively small values of \(f(x)\). But as \(x\) increases, the values of \(f(x)\) grow larger, albeit at a rate slower than exponential functions.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. This results in rapid growth or decay, depending on the base of the exponent.
The function \( g(x) = 3^x \) is a classic example of an exponential function.

• The distinguishing feature of exponential functions is their rapid rate of change, causing the value of the function to increase or decrease exponentially.
• In \( g(x) \), as \(x\) increases by one unit, \(g(x)\) is multiplied by 3 each time.

This multiplicative growth leads to exponential functions surpassing polynomial functions rapidly with larger values of \(x\).
Initially, exponential functions may start below or near polynomial functions, but their growth rate dramatically changes the scenario as \(x\) gets larger.
In our comparison, \( g(x) \) quickly overtakes \( f(x) \) by the time \(x=7\). This exemplifies the power of exponential growth.

Graphing Functions

Graphing functions allows for a visual comparison and analysis of different types of functions. This can help determine where they intersect, their relative growth, and behavior as \(x\) becomes very large or very small.

• For the comparison between \( f(x) = x^3 \) and \( g(x) = 3^x \), plotting both on the same set of axes gives insight into how these functions behave.
• Initially, \( f(x) \) starts with lower values while \( g(x) \) starts slightly higher at \( x = 0 \).

By the time both functions reach \( x = 6 \), they are equal at 729.
By \( x = 7 \), \( g(x) \) takes a dramatic lead, illustrating the exponential function's rapid growth.
Graphing functions not only helps in visualizing the functional form but also in making intuitive sense of numerical calculations and differential growth rates of polynomial versus exponential functions.