Question

Group Activity End Behavior Models Suppose that \(g_{1}(x)\) is a right end behavior model for \(f_{1}(x)\) and that \(g_{2}(x)\) is a right end behavior model for \(f_{2}(x) .\) Explain why this makes \(g_{1}(x) / g_{2}(x)\) a right end behavior model for \(f_{1}(x) / f_{2}(x)\)

Short answer

The division of the end behaviour models \(g_{1}(x) / g_{2}(x)\) gives the right end behaviour model for the quotient of the functions \(f_{1}(x) / f_{2}(x)\). This is because the end behavior models accurately capture the behavior of the respective functions as 'x' approaches positive infinity, so their division will reflect the end behavior of the division of the original functions.

Step-by-step solution

Understand End Behaviour Models

The end behavior model of a function describes the behavior of the function as the input 'x' approaches positive or negative infinity. The right end behavior model, denoted as \(g(x)\), describes the function's behavior as 'x' approaches positive infinity. Here, we have two functions \(f_{1}(x)\) and \(f_{2}(x)\) with their right end behavior models represented by \(g_{1}(x)\) and \(g_{2}(x)\) respectively.

Explain Division of End Behaviour Models

Dividing \(g_{1}(x)\) by \(g_{2}(x)\) results in a new function. It is important to note here, when \(x\) approaches \(+\infty\), the behavior of \(g_{1}(x) / g_{2}(x)\) will reflect the behavior of the original functions divided, that is, \(f_{1}(x) / f_{2}(x)\). This is because the end behavior models \(g_{1}(x)\) and \(g_{2}(x)\) accurately reflect the behavior of the functions \(f_{1}(x)\) and \(f_{2}(x)\) as \(x\) approaches \(+\infty\). Therefore, the division of these end behavior models will mimic the division of the original functions.

Key concepts

Limit of a Function

Understanding the concept of a function's limit is foundational when studying calculus and higher-level mathematics. A limit attempts to find the value that a function approaches as the input, or 'x', gets closer to a certain number. This number can often be positive infinity () or negative infinity (), both of which are not actual numbers but rather concepts representing unbounded growth or decrease.

The key here is that we're not interested in the function's value at the point of interest, but rather the value it 'approaches' as it gets infinitely close to that point. So, in the context of our exercise, when evaluating the end behavior models for the functions and as 'x' approaches positive infinity, we're essentially looking at the limit of towards positive infinity () and doing the same for the limit of towards positive infinity.

Asymptotic Behavior

When we discuss the asymptotic behavior of a function, we are referring to how the function behaves as it moves closer to a line that it will never actually reach, known as an asymptote. These asymptotes can be horizontal, vertical or even oblique. In terms of horizontal asymptotes, which are common when 'x' approaches infinity, they help us understand the end behavior of a function. It gives us a 'ceiling' or 'floor' towards which the function values will tend, but never cross.

With and , their respective end behavior models and essentially serve as tools to analyze how each function behaves in the long run, or asymptotically, as 'x' goes to positive infinity. If grows without bounds, could serve as a horizontal asymptote guiding the overall shape of the graph of at extreme values of 'x'.

Rational Functions Analysis

Rational functions — ratios of polynomials — exhibit various types of end behavior, often determined by the degrees of the polynomials in the numerator and the denominator. When analyzing rational functions, one typically looks at the degrees of these polynomials to predict their behavior for very large or very small values of 'x'.

In our exercise, the end behavior model of divided by can be seen as a simplification that predicts the behavior of the more complex function at extreme values. Assuming both and represent the dominant behavior of their corresponding functions, their ratio would capture the essence of how the quotient of the original functions behaves as 'x' tends towards infinity. The appropriateness of using —as a representative model to describe the end behavior of —speaks to the power of using simpler functions to grasp the behavior of more complex ones over an unbounded domain.