Question

Graph the functions \(f(x)=6^{x}\) and \(g(x)=x^{6}\) in the same viewing window. Where do these graphs intersect? As \(x\) increases, which function grows more rapidly?

Short answer

By graphing the functions, it should be observable that functions \(f(x) = 6^x\) and \(g(x) = x^6\) intersect at two points, \(x = 0\) and \(x = 1\). As \(x\) increases, the exponential function \(f(x) = 6^x\) grows more rapidly.

Step-by-step solution

Graph functions

Begin by plotting both functions \(f(x) = 6^x\) and \(g(x) = x^6\) in the same graph, which will make it easier to compare them and find their intersection points.

Find Intersection

The intersection points are where both functions are equal, i.e., \(f(x) = g(x)\). This results in the equation \(6^x = x^6\). Solving this equation will provide the \(x\) values of intersections.

Analyzing Growth Rates

After plotting the functions, observe as \(x\) increases which function grows more rapidly. Based on the nature of these functions, the function \(f(x) = 6^x\) (exponential function) is expected to grow more quickly than \(g(x) = x^6\) (polynomial function) as \(x\) increases.

Key concepts

Exponential Functions

Exponential functions can initially seem less intuitive compared to linear or polynomial functions, but they represent some of the most dynamic and fastest-growing phenomena in both mathematics and the real world. An exponential function is defined by the formula f(x) = a^x, where a is a constant base that is greater than 0, and x is the exponent. As x increases, the value of f(x) increases exponentially, not at a constant rate but at a rate proportional to its current value, which leads to a signature steep curve as seen in the function f(x) = 6^x.

This kind of growth makes exponential functions particularly important in areas like compound interest, population growth, and radioactive decay. In each of these, the rate of growth or decay is tied to the current amount, which leads to explosive increase or decrease over time.

Polynomial Functions

On the other hand, polynomial functions represent a sum of powers of the variable x and are one of the most common types of functions in mathematics. They are expressed in the form p(x) = a_n x^n + ... + a_2 x^2 + a_1 x + a_0, where a_n, a_2,... are coefficients and n is a non-negative integer called the degree of the polynomial. The biggest power in the polynomial determines its end behavior and roughly how many turns the graph will have.

For example, the function g(x) = x^6 from our problem is a polynomial function of degree 6, which means it has a symmetric, smooth, and somewhat predictable curve. Polynomial functions like this show up in physics, economics, and even the path of projectiles. They grow fast, but not as fast as exponential functions, especially as the value of x becomes very large.

Intersection of Graphs

Finding the Point of Intersection

When graphing two functions on the same set of axes, the intersection points hold significant importance because they represent the values at which the two functions equal each other. To find these points algebraically, you set the equations of the two functions equal to each other and solve for the variable. In our exercise, by setting 6^x equal to x^6, we're searching for the value(s) of x where both the exponential and the polynomial function output the same result.

Understanding intersections is vital in real-world scenarios such as determining the break-even point in a business model or when two different growth patterns will yield the same value, among other examples.

Comparing Growth Rates

Exponential vs. Polynomial Growth

Growth rates of functions are a foundational concept in understanding how different mathematical models predict changes over time. Exponential growth, as represented by the function 6^x, is characterized by a growth rate that increases over time. The greater the value of x, the steeper the incline of the graph. In stark contrast, polynomial growth, represented by the function x^6, increases at a rate determined by its highest degree term, but this increase is far less dramatic than that of an exponential function.

In our exercise, while the polynomial function starts off growing rapidly, over the long run, exponential functions like 6^x will far outpace it. This reflects patterns seen in finance, biology, and various other fields, where understanding the nature of growth rates of different functions is crucial for making predictions and informed decisions.