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Chapter 2: Problem 48

Explain why the composition of two rational functions is a rational function.

Short Answer

Expert verified
The composition of two rational functions, f(x) and g(x), is a rational function because when we compose them as (f鈭榞)(x) = f(g(x)), we obtain an expression in the form \(\frac{P(x)}{Q(x)}\), where P(x) and Q(x) are polynomials in x. This confirms that the composition of two rational functions is indeed a rational function.

Step by step solution

01

Definition of a Rational Function

A rational function is a function that can be represented as the quotient of two polynomials, say P(x) and Q(x), where P(x) and Q(x) are polynomials with real coefficients, and Q(x) is not the zero polynomial. Mathematically, a rational function can be represented as: \[f(x) = \frac{P(x)}{Q(x)}\]
02

Composition of Two Rational Functions

Let's consider two rational functions, f(x) and g(x), such that: \[f(x) = \frac{P_1(x)}{Q_1(x)}\hspace{1cm} \text{and} \hspace{1cm} g(x) = \frac{P_2(x)}{Q_2(x)}\] The composition of the two functions, denoted as (f鈭榞)(x), is given by: \[(f \circ g)(x) = f(g(x))\]
03

Substituting the Rational Functions in the Composition

Now, substitute f(x) and g(x) in the composition (f鈭榞)(x): \[(f \circ g)(x) = f(g(x)) = f\left(\frac{P_2(x)}{Q_2(x)}\right)\] Next, substitute the expression for f(x) into the composition: \[(f \circ g)(x) = \frac{P_1\left(\frac{P_2(x)}{Q_2(x)}\right)}{Q_1\left(\frac{P_2(x)}{Q_2(x)}\right)}\]
04

Simplification of the Composition

We can simplify the expression using a common denominator, which will give us the composition in the form of a rational function. The common denominator is \(Q_1\left(\frac{P_2(x)}{Q_2(x)}\right) * Q_2(x)\): \[(f \circ g)(x) = \frac{P_1\left(\frac{P_2(x)}{Q_2(x)}\right) * Q_2(x)}{Q_1\left(\frac{P_2(x)}{Q_2(x)}\right) * Q_2(x)}\]
05

Result as a Rational Function

Notice that the numerator and the denominator are polynomials of x. Let P(x) and Q(x) be those polynomials: \[P(x) = P_1\left(\frac{P_2(x)}{Q_2(x)}\right) * Q_2(x)\] \[Q(x) = Q_1\left(\frac{P_2(x)}{Q_2(x)}\right) * Q_2(x)\] So we can write the composition (f鈭榞)(x) as the quotient of two polynomials, confirming that the composition of two rational functions is indeed a rational function: \[(f \circ g)(x) = \frac{P(x)}{Q(x)}\]

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