Problem 45 - Calculus for Biology and Medicine

Question

Let $$ f(x)=\frac{x^{2}}{4+x^{2}}, \quad x \geq 0 $$ (a) Use a graphing calculator to graph \(f(x)\). (b) On the basis of your graph in (a), find the range of \(f(x)\). (c) What happens to \(f(x)\) as \(x\) gets larger?

Step-by-Step Solution

Answer

The range of \( f(x) \) is \([0, 1)\). As \( x \) increases, \( f(x) \) approaches 1.

Step 1: Understanding the Function

The given function is \( f(x) = \frac{x^2}{4+x^2} \) and is defined for \( x \geq 0 \). This means that we are looking at the function's behavior only for non-negative values of \( x \).

Step 2: Analyzing the Graph

To graph \( f(x) \), input the function \( f(x) = \frac{x^2}{4+x^2} \) into a graphing calculator or software, adjusting the window to allow for sufficient visualization of the function as \( x \) ranges from 0 to a larger number.

Step 3: Observing Graph Behavior

Notice that as you graph \( f(x) \), the curve starts at 0 when \( x=0 \) and approaches 1 as \( x \) becomes very large. This suggests that the function increases from 0 to approach 1.

Step 4: Identifying the Range

Based on the graph, as \( x \) varies from 0 to positive infinity, \( f(x) \) begins at 0 and asymptotically approaches 1. Therefore, the range of \( f(x) \) is \( [0, 1) \).

Step 5: Analyzing Limit Behavior

As \( x \rightarrow \infty \), \( f(x) = \frac{x^2}{4+x^2} \rightarrow 1 \) because the \( x^2 \) term dominates both the numerator and the denominator, making the fraction approach 1. This is called the horizontal asymptote of the function.

Key Concepts

Function Behavior

Understanding how a function behaves is pivotal in grasping its nature and predicting its output under various circumstances. When considering the function \(f(x) = \frac{x^2}{4+x^2}\) for \(x \geq 0\), it is crucial to examine how the output of the function changes as the input \(x\) increases.
Initially, when \(x = 0\), the function returns 0 since \(f(0) = \frac{0}{4+0} = 0\). As \(x\) increases, both the numerator and denominator grow, but the rate of growth here is significant.\(x^2\) grows faster compared to the constant 4 added in the denominator. This results in the function increasing.
Visualizing this growth with a graph further confirms that the curve rises from 0 and gradually flattens out towards 1. Such insights into function behavior are essential for predicting its range and limits.

Range of a Function

A function's range is the set of all possible output values it can generate. For the function \(f(x) = \frac{x^2}{4+x^2}\), determining the range involves examining the graph and using logical reasoning.
Upon graphing the function, we notice that it starts from a point \((0,0)\). As \(x\) increases, the value of \(f(x)\) heads towards, but never quite reaches, 1. Hence, the range doesn't include 1 itself.

• At \(x = 0\), \(f(x) = 0\).
• As \(x \to \infty\), \(f(x)\) approaches 1 without ever achieving it.

This indicates that, for the values of \(x\) restricted to non-negative numbers, \(f(x)\) takes every value from 0 up to, but not including, 1. Mathematically, it's expressed as the interval \([0, 1)\). This interval confirms that while the function can get arbitrarily close to 1, it remains slightly below it.

Limit Analysis

Limit analysis deals with understanding the behavior of a function as the input approaches a particular value. In our function, \(f(x) = \frac{x^2}{4+x^2}\), we need to analyze what happens when \(x\) approaches infinity.
As \(x\) becomes very large, the \(x^2\) in both the numerator and the denominator plays a pivotal role. The function simplifies significantly during this analysis:

• The numerator becomes approximately \(x^2\).
• The denominator becomes approximately \(x^2\) (since 4 becomes negligible compared to \(x^2\)).

Thus, the function simplifies to \(\frac{x^2}{x^2} = 1\) as \(x\) approaches infinity. However, \(f(x)\) never actually reaches 1. Instead, it gets continually closer as \(x\) swells.
This leads us to conclude that \(y = 1\) is a horizontal asymptote for \(f(x)\). This concept of a limit provides a powerful tool for predicting behavior without requiring direct computation of function values for extremely large inputs.