Problem 98 - Calculus Volume 1

Question

For the following exercises, determine whether the statement is true or false. Explain why. \(f(x)=(4 x+1) /(7 x-2) \quad\) is a transcendental function.

Step-by-Step Solution

Answer

False, because it is a rational, not a transcendental function.

Step 1: Understand Transcendental Functions

A transcendental function is a function that is not algebraic. Typical transcendental functions include exponential functions, logarithms, trigonometric functions, and some special functions. On the other hand, an algebraic function is one that can be expressed using a finite number of algebraic operations (addition, subtraction, multiplication, division, and taking roots) on polynomials.

Step 2: Analyze the Given Function

Given the function \( f(x) = \frac{4x+1}{7x-2} \), observe that both the numerator (\(4x+1\)) and the denominator (\(7x-2\)) are polynomials. The function itself is expressed as a ratio of two polynomials.

Step 3: Determine the Nature of the Function

Since \( f(x) \) is a ratio of two linear polynomials, it is a rational function. Rational functions are a subset of algebraic functions because they can be expressed using a finite number of algebraic operations on polynomials.

Step 4: Conclusion

Since \( f(x) \) is a rational function, it cannot be transcendental. Therefore, the statement that \( f(x) \) is a transcendental function is false.

Key Concepts

Algebraic Functions

Algebraic functions are fundamental in mathematics and include any function that can be formed using a finite number of arithmetic operations on polynomials. These operations include addition, subtraction, multiplication, division, and taking roots. If you can express a function using these operations on polynomials, it qualifies as algebraic. For instance,

• Consider the function: \( h(x) = \sqrt{x^2 + 7} \). This is an algebraic function because it involves a square root of a polynomial.
• Functions like \( g(x) = x^3 - 4x^2 + 6 \) are also algebraic because they are polynomial expressions.

However, not all functions dealing with arithmetic operations are algebraic, for example, transcendental functions like exponentials and trigonometric functions don't fit the criteria as they cannot be expressed through a finite combination of polynomials. Understanding the nature of these functions helps in defining whether they are algebraic or not.

Rational Functions

Rational functions form an important class of algebraic functions. They are literally ratios of two polynomials. If you can express a function as the quotient of two polynomial functions, then it's rational. An example is the function given in the original exercise:

• \( f(x) = \frac{4x+1}{7x-2} \) is a rational function since both the numerator \(4x + 1\) and the denominator \(7x - 2\) are polynomials.

Rational functions can model real-life situations where one quantity is dependent on another, and they have vertical and horizontal asymptotes based on their denominators and overall exponents. Note that rational functions are all algebraic functions but crucially, not all algebraic functions are rational. This distinction is nuanced but essential.

Polynomials

Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are very versatile and form the building blocks for both algebraic and rational functions. A polynomial looks like \( p(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \(a_n, a_{n-1}, \ldots, a_0\) are coefficients and \( n \) is a non-negative integer.

• For example, the expression \( q(x) = 3x^3 - 2x + 5 \) is a polynomial of degree 3.

Polynomials can be added, subtracted, multiplied, and even divided, although division may lead to rational functions rather than polynomials. They form the basis of algebraic functions, meaning any algebraic function can be broken down into operations involving polynomials. Their simplicity in terms of operations starkly contrasts with transcendental functions, which cannot be reduced to polynomial form.