Question

Which of these two functions will increase faster as the value of \(x\) becomes larger? Function \(A: f(x)=\frac{x^3}{x-2}\) Function B: $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 20 & 50 & 100 \\ \hline \boldsymbol{f ( x )} & 4 & 6 & 8 & 10 & 12 & 44 & 104 & 204 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|} \hline \text { Select... } & \boldsymbol{\nabla} \\ \hline \text { Function A } \\ \hline \text { Function B } \\ \hline \end{array} $$

Short answer

Function A increases faster as \( x \) becomes larger.

Step-by-step solution

Understand Functions

Function A is given by the expression \[ f(x) = \frac{x^3}{x-2} \] For Function B, values of \( f(x) \) are given in a table for specific values of \(x\).

Analyze Function A

To understand how Function A behaves as \(x\) becomes larger, we can look at its leading term since the polynomial numerator grows faster than the linear denominator. When \(x\) is very large, \[ f(x) \approx \frac{x^3}{x} = x^2 \] which grows very rapidly.

Calculate Growth Rate of Function B

Review the values given for Function B in the table by dividing the differences of \( f(x) \) values by the differences of \( x \) values. For example, from \( x = 2 \) to \( x = 3 \): \[ \text{Change in } f(x) = 10-8 = 2 \; \text{and} \; \text{Change in } x = 3-2 = 1 \] hence, \( \frac{\text{Change in } f(x)}{\text{Change in } x} = 2 \)

Compare Growth Rates

The growth rate of Function A is polynomial as it approximates \( x^2 \) for large \( x \), indicating rapid increase. Function B increases linearly, so it grows significantly slower than a polynomial function. Therefore, Function A will increase faster as \( x \) becomes larger.

Key concepts

polynomial functions

Polynomial functions are expressions that involve sums of powers of the variable, such as \(x\). The general form of a polynomial function is \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \] where \(a_n, a_{n-1}, \dots, a_0\) are coefficients and \( n \) is a non-negative integer. In the given exercise, Function A is a polynomial function. It is provided as \[ f(x) = \frac{x^3}{x-2} \].

To simplify and understand its growth, as \( x \) gets very large, the term \( -2 \) in the denominator becomes negligible. This allows us to approximate the function as \[ f(x) \approx \frac{x^3}{x} = x^2 \], which is a polynomial of degree 2. The highest power term, \(x^2 \) in this case, dominates the growth of the function.

Polynomials grow rapidly, especially as the highest degree term \(x^n\) with \(n \geq 2\) tends to infinity. This is why Function A, which approximates to \(x^2,\) grows much faster compared to linear functions.

linear functions

Linear functions are a fundamental concept in mathematics. They can be described by the equation \[ f(x) = mx + b \], where \( m \) is the slope and \( b \) is the y-intercept. In simple terms, linear functions produce a straight line when graphed.

In the exercise, Function B can be seen as approximating a linear function in behavior. By analyzing the table given for Function B, we see a consistent increase corresponding to specific increments of \(x\). For instance, when the change in \( x \) values from 2 to 3 results in a consistent change in \( f(x) \) values, \[ \text{Change in } f(x) = 10-8 = 2 \] This constant rate of increment means Function B can be described by a linear pattern.

Linear functions grow at a constant rate. This steady, predictable growth is in stark contrast to polynomial functions, which can grow at an accelerating rate due to higher exponent terms.

growth rate analysis

Growth rate analysis entails comparing how quickly functions increase as the variable \( x \) becomes larger. When analyzing the growth rates of different functions, it's essential to look at the leading terms and how they behave.

For Function A, which is approximated as \( x^2 \), the growth is polynomial. Polynomial growth refers to the expansion where the rate of increase accelerates as \( x \) becomes large. The \( x^2 \) term indicates that every time \( x \) doubles, \( f(x)\) quadruples, showing a rapid rate of increase.

In contrast, Function B has a linear growth rate, evidenced by consistent and equal increments in \( f(x) \) given equal increments in \( x \). For example, when \( x \) increases from 3 to 4, \( f(x) \) increases by 2 just as it did from 2 to 3.

Comparing these growth rates, polynomial functions like Function A grow much more rapidly than linear functions like Function B. Therefore, Functions with higher polynomial degrees will always outpace linear functions as \( x \) becomes very large.