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When analyzing the end behavior of functions, we are essentially examining how the function behaves as the variable x approaches very large or very small values. This is crucial for understanding which function grows faster and will eventually dominate. In our given problem, we compare the functions:

\[ f(x) = 100 + 500x \] and \[ g(x) = 2(1.005)^x \]

The first function, \( f(x) \), is linear, which means it grows at a steady rate as x increases. The second function, \( g(x) \), is exponential. Exponential functions generally grow much faster than linear functions as x becomes large.

Therefore, even a small base such as 1.005, when raised to a large power, results in a much larger value than the linear component. Thus, as \( x \rightarrow +\infty \), \( g(x) \) will dominate \( f(x) \).

Solving inequalities involves finding the range of values for which one function is greater than another. In our case, we need to determine when \( g(x) \) is greater than \( f(x) \). This translates to solving the inequality:

\[ 2(1.005)^x > 100 + 500x \]

To tackle this, we rearrange it into:

\[ 2(1.005)^x - 100 - 500x > 0 \]

Inequalities involving exponential and linear terms can be challenging to solve algebraically. Thus, graphical methods are often used. By plotting the graphs of the two functions, we can visually determine the x-values where \( g(x) \) exceeds \( f(x) \). This graphical method provides a straightforward way to see where the inequality holds true.

Graphical analysis is a powerful tool for solving equations and inequalities, and for understanding the behavior of functions. When we graph \( f(x) = 100 + 500x \) and \( g(x) = 2(1.005)^x \), we look for their intersection points. Where \( g(x) \) is above \( f(x) \), we have \( g(x) > f(x) \).

Using a graphing calculator or a graphing program, plot both functions. The point where the graphs intersect is the solution to \( 2(1.005)^x = 100 + 500x \). The values of x to the right of this intersection point indicate where \( g(x) > f(x) \).

For instance, if they intersect at \( x_0 \), the interval where \( g(x) \) dominates \( f(x) \) would be \( (x_0, +\infty) \). This graphical intersection helps us understand the relationship between the two functions visually and numerically.

Function dominance refers to which function grows faster as x approaches infinity. In our analysis, we observed that the exponential function \( g(x) \) will eventually dominate the linear function \( f(x) \). This is a common scenario where exponential growth outpaces linear growth due to the exponential rate of increase.

Linear functions, represented generally by \( ax + b \), increase at a constant rate of a units per increase in x. Exponential functions, noted as \( a(b)^x \), increase at a rate proportional to their current value, leading to much faster growth as x increases.

In our problem, \( g(x) \)'s base of 1.005, although small, ensures exponential growth. As \( x \rightarrow +\infty \), the multiplicative effect of repeatedly multiplying by 1.005 results in very large values, causing \( g(x) \) to surpass and dominate \( f(x) \) even for reasonably large x.