Problem 45 - Brief 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.) $$ f(x)=x^{2}+x^{1 / 2} $$

Step-by-Step Solution

Answer

The function is none of these.

Step 1: Define a Polynomial Function

A polynomial function is a function that can be expressed in the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( n \) is a non-negative integer and \( a_i \) are constants. The exponents of \( x \) must be integers.

Step 2: Analyze the Given Function

The given function is \( f(x) = x^2 + x^{1/2} \). Here, \( x^2 \) fits the polynomial definition with an integer exponent. However, \( x^{1/2} \) has a fractional exponent.

Step 3: Determine Function Type

Since \( x^{1/2} \) has a fractional exponent, the function cannot be a polynomial. It's also not a rational function, exponential function, or piecewise linear function based on its form.

Step 4: Conclusion

Given the fractional exponent, the function \( f(x) = x^2 + x^{1/2} \) is not a polynomial, rational, exponential, or piecewise linear function. Therefore, it is classified as 'none of these.'

Key Concepts

Polynomial Functions

Polynomial functions are one of the most fundamental types of functions in mathematics. They are defined by expressions that include terms with non-negative integer exponents of a variable. For example,

• \( f(x) = x^2 + 2x + 1 \)
• \( g(x) = 4x^3 - x + 5 \)

Each term in a polynomial is a product of a constant coefficient and a power of the variable. Notice that every exponent here is a whole number, which is crucial for the equation to be considered a polynomial.
Polynomials are crucial for modeling a wide range of real-world situations and form the building blocks of more complex functions.
Within a polynomial, the degree of the function is determined by the highest power of the variable present. This degree can tell us a lot about the function's shape and how its graph behaves.

Rational Functions

Rational functions represent the quotient of two polynomials. These functions take the form \( R(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \).

• An example of a rational function is \( R(x) = \frac{x^2 + 1}{x - 3} \)
• Another example could be \( R(x) = \frac{x^2 - 4}{x^2 + x + 1} \)

Rational functions are notable for their potential to create asymptotes in graphs, which means that as \( x \) approaches certain values, the function's value increases or decreases without bound.
They can model scenarios where you have a ratio relationship, often appearing in problems involving rates, mixtures, and proportional relations.

Exponential Functions

Exponential functions involve a constant raised to a variable exponent, represented as \( f(x) = a^x \) where \( a \) is a positive constant. These functions are characterized by rapid growth or decay.
Examples include:

• \( f(x) = 3^x \): A rapidly growing exponential function.
• \( f(x) = (1/2)^x \): An example of an exponential decay, where the function decreases as \( x \) increases.

Exponential functions are widely used to describe processes that change at a rate proportional to their current value, such as population growth, radioactive decay, and interest calculations.
Their graphs are smooth curves that, unlike polynomials, never touch the horizontal axis, emphasizing the continual increase or decrease.

Piecewise Linear Functions

Piecewise linear functions are composed of multiple linear segments that are applied over different intervals of the independent variable. These functions are often expressed with different formulas for each interval.

• For example, \( f(x) \) might be defined as:
  • \( f(x) = 2x + 3 \) if \( x < 0 \)
  • \( f(x) = -x + 1 \) if \( x \geq 0 \)
  This creates a graph consisting of straight lines that break or change slope at specified points.
  Piecewise linear functions are very useful in modeling situations where there is a sudden change in condition or behavior, such as tax rate brackets or stepped billing rates. They simplify complex scenarios by breaking them down into simple linear interactions.
  Understanding these functions requires identifying each linear segment and its applicable domain.