Problem 44 - Applied Calculus

Question

Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=x^{3}-x^{2 / 3} $$

Step-by-Step Solution

Answer

The function is none of these.

Step 1: Analyze the Terms of the Function

Inspect the expression \(f(x) = x^3 - x^{2/3}\). Identify the nature of each term with respect to the definition of polynomial functions.

Step 2: Definition of Polynomial Function

A polynomial function is a function of the form \(a_n x^n + a_{n-1} x^{n-1} + \.\.\. + a_1 x + a_0\), where the exponents are non-negative integers.

Step 3: Apply Polynomial Characteristics to the Function

Check if the exponents in \(f(x) = x^3 - x^{2/3}\) fit the characteristics of a polynomial. \(x^3\) fits, as its exponent is a non-negative integer. However, \(x^{2/3}\) has an exponent which is a fraction, not a non-negative integer.

Step 4: Conclusion Based on Characteristics

Since not all the terms in \(f(x)\) have non-negative integer exponents, \(f(x)\) cannot be classified as a polynomial function. This eliminates \(f(x)\) as a polynomial.

Step 5: Consider Other Types of Functions

Evaluate if the function fits other categories like rational, exponential, or piecewise linear. However, it's important to note that rational functions involve ratios of polynomials, exponential functions have constant bases with variable exponents, and piecewise functions involve different expressions based on intervals.

Step 6: Determine the Function's Type

Since \(f(x) = x^3 - x^{2/3}\) does not satisfy the definitions for polynomial, rational, exponential, or piecewise linear functions, it falls into the category of 'none of these.'

Key Concepts

Polynomial Function

In mathematics, a polynomial function is a special kind of function. It is composed of terms that involve variables raised to whole number powers. A general polynomial function looks like this:

• Expression: \(a_n x^n + a_{n-1} x^{n-1} + \.\.\. + a_1 x + a_0\)
• The exponents \(n, n-1,...,1,0\) are all non-negative integers.

This type of function is straightforward and only consists of whole amounts in the exponents. The simplest example is a linear function, like \(f(x) = x\), which is a first-degree polynomial. A quadratic function such as \(f(x) = x^2 + 3x + 1\) is a second-degree polynomial.
Polynomials are quite versatile and appear in various applications, including physics and economics, due to their easy-to-understand structure and properties, making them central in calculus and algebra. The function, \(f(x) = x^3 - x^{2/3}\), as noted in the analysis, cannot be a polynomial function because it contains \(x^{2/3}\), which has a fractional exponent, thus violating the polynomial criteria.

Rational Function

A rational function is defined as the ratio of two polynomial functions. Essentially, it looks like a fraction where both the numerator and the denominator are polynomials. The general form of a rational function is:

• Expression: \(\frac{P(x)}{Q(x)}\)
• Both \(P(x)\) and \(Q(x)\) are polynomials.

Rational functions can be more complex because the behavior near the zeros of the denominator can lead to vertical asymptotes or undefined points. For instance, the function \(f(x) = \frac{x^2 + 1}{x - 3}\) is a rational function.
Unlike a polynomial function, a rational function can have discontinuities. However, the function \(f(x) = x^3 - x^{2/3}\) does not meet the criteria of a rational function since it is not expressed as a ratio of polynomials. Instead, it is a single expression with a non-integer exponent. Hence, it does not qualify as a rational function.

Exponential Function

An exponential function is distinguished by a constant base raised to a variable exponent. This form is represented as:

• Expression: \(a^x\)
• Where \(a\) is a constant, and \(x\) is a variable.

Some common examples include \(f(x) = 2^x\) or \(f(x) = e^x\), where \(e\) is the mathematical constant approximately equal to 2.71828.
Exponential functions are key in modeling situations involving exponential growth or decay, such as population growth, radioactive decay, or interest compounded continuously. These functions are characterized by rapid increases or decreases; they are not influenced by fractional powers or additional terms. The function in question, \(f(x) = x^3 - x^{2/3}\), does not fit the exponential function category due to the absence of a constant base and the presence of a variable raised to a fractional power.

Piecewise Linear Function

Piecewise linear functions are types of functions defined by different linear expressions over different intervals of the domain. They "piece together" different linear functions to cover the domain of interest.

• This kind of function is represented by different equations depending on the interval of the input value.
• Typically, these functions have distinct linear fragments, which means they often include functions like \(f(x) = 2x\) for \(x < 3\) and \(f(x) = x + 4\) for \(x \geq 3\).

Such functions are useful in scenarios where different rules apply across different ranges. Unlike the continuous nature of a polynomial function, piecewise functions can have abrupt changes at the boundaries of intervals.
The function \(f(x) = x^3 - x^{2/3}\) is not a piecewise linear function since it is composed of terms with power that involve fractional powers and do not switch definitions based on different intervals of \(x\). It maintains a consistent form across its domain rather than changing behavior over intervals.