Problem 22 - Thomas Calculus

Question

Find the range of $$y=2+\frac{x^{2}}{x^{2}+4}$$

Step-by-Step Solution

Answer

The range of \( y = 2 + \frac{x^2}{x^2 + 4} \) is \([2, 3)\).

Step 1: Understanding the Function

The given function is \( y = 2 + \frac{x^2}{x^2 + 4} \). Note that the term \( \frac{x^2}{x^2 + 4} \) is a rational function where \( x^2 \) is a non-negative term, and the denominator \( x^2 + 4 \) ensures that it is always positive. Thus, \( \frac{x^2}{x^2 + 4} \) must be a fraction between 0 and 1, but not including 1.

Step 2: Finding the Bounds for the Fraction

Consider the expression \( \frac{x^2}{x^2+4} \). For every real number \( x \), the term \( x^2 \) is non-negative, and by algebraic manipulation, \( 0 \leq \frac{x^2}{x^2+4} < 1 \). As \( x \) approaches infinity, the term approaches 1, but never reaches it.

Step 3: Determining the Range of Original Function

The value of \( y \) is determined by adding 2 to \( \frac{x^2}{x^2+4} \). Thus, the bounds of the function \( y = 2 + \frac{x^2}{x^2 + 4} \) can be deduced: \( 2 \leq y < 3 \). Hence, the range of \( y \) is all values from 2 (inclusive) to 3 (non-inclusive).

Key Concepts

Rational Functions

Rational functions are expressions that involve a ratio of two polynomials. In simpler terms, they are functions made by dividing one polynomial by another. In the case of the function \( y = 2 + \frac{x^2}{x^2 + 4} \), the term \( \frac{x^2}{x^2+4} \) is the rational part. Rational functions can behave in complex ways due to the interplay of the numerator and the denominator. However, the general behavior of a rational function is determined by the degrees of these two polynomials.Here, the numerator and denominator both have the same degree (2, since \( x^2 \) is the highest term), which means the rational part approaches a constant value as \( x \) gets very large (in this case 1). One interesting property of rational functions is their range, coming from how the values of the numerator and denominator interact, providing insight into the values that the function can take. This insight helps in determining how the overall function behaves as \( x \) becomes large or very small.

Inequalities

Inequalities are mathematical expressions describing the relationship of one value being larger or smaller than another. In analyzing functions, inequalities help define the limits within which a function operates. For the function \( y = 2 + \frac{x^2}{x^2+4} \), the rational part \( \frac{x^2}{x^2+4} \) stays within certain bounds due to the nature of \( x^2 \) and \( x^2 + 4 \).

• Since \( x^2 \) is always non-negative, its smallest value is 0 when \( x = 0 \).
• The value of the rational function \( \frac{x^2}{x^2+4} \) thus starts at 0 when \( x = 0 \) and comes extremely close to, but never actually reaching, 1 as \( x \) grows larger.

Understanding inequalities is crucial for determining the range of possible values \( y \) can take. Inequalities like \( 0 \leq \frac{x^2}{x^2+4} < 1 \) contribute to understanding how \( y \) varies from one point to another, especially the boundaries forming the minimum and maximum values \( y \) can achieve.

Function Approaches Infinity

When we say a function 'approaches infinity,' we refer to how its value behaves as the input (\( x \)) becomes very large. For the function \( y = 2 + \frac{x^2}{x^2 + 4} \), examining behavior as \( x \rightarrow \infty \) is fascinating.A helpful approach is to substitute large values for \( x \) and observe the effect on \( y \). Here, as \( x \) tends to infinity:

• The function \( \frac{x^2}{x^2 + 4} \) tends towards 1. This is because the \(+4\) in the denominator becomes insignificant against the \( x^2 \), making the ratio roughly 1.
• Therefore, \( y = 2 + \frac{x^2}{x^2+4} \) approaches 3, but never quite reaches it.

This behavior, called a limit, helps in determining the range, allowing us to conclude that \( y < 3 \). Recognizing how a function behaves as \( x \) becomes extremely large or small is crucial in understanding its complete behavior, particularly in identifying any asymptotic tendencies, like approaching but not reaching certain values.