Problem 35 - 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.) $$ f(x)=5^{x} $$

Step-by-Step Solution

Answer

The function \( f(x) = 5^x \) is an exponential function.

Step 1: Understanding Polynomial Functions

Polynomial functions are defined by the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \) where \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer. In our function \( f(x) = 5^x \), the powers of \( x \) are not constant integers, which does not qualify it as a polynomial.

Step 2: Understanding Rational Functions

Rational functions are functions of the form \( \frac{p(x)}{q(x)} \) where both \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) eq 0 \). The function \( f(x) = 5^x \) does not involve any polynomial divided by another polynomial. Therefore, it is not a rational function.

Step 3: Understanding Exponential Functions

Exponential functions are functions of the form \( f(x) = a^x \) where \( a \) is a constant and \( a > 0 \). Since our function \( f(x)=5^x \) fits this description with \( a = 5 \), it is classified as an exponential function.

Step 4: Understanding Piecewise Linear Functions

Piecewise linear functions are defined by different linear equations over various intervals of the domain. Since \( f(x)=5^x \) is not composed of different linear equations applied to intervals, it is not a piecewise linear function.

Key Concepts

Exponential Functions

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. These functions can generally be written in the form \( f(x) = a^x \), where \( a \) is a positive constant greater than zero, and \( x \) is the exponent. In the function \( 5^x \), 5 is the base, and \( x \) is the exponent.

Exponential functions are essential in modeling growth and decay processes, such as population growth, radioactive decay, and compound interest. They have unique properties, such as:

• They are always positive for real-numbered exponents.
• If the base \( a > 1 \), the function grows as \( x \) increases.
• If \( 0 < a < 1 \), the function decreases as \( x \) increases.

Polynomial Functions

Polynomial functions are expressions that consist of variables and coefficients, organized in terms of non-negative integer powers of the variable. They have the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0 \). Here, \( a_n, a_{n-1}, \ldots, a_0 \) are constants with \( n \) as a non-negative integer.

Let's consider a simple example: \( f(x) = 3x^3 + 2x^2 + x + 7 \) is a polynomial of degree 3.

Polynomials are essential in various fields due to their computable nature. Key characteristics include:

• Their domain is all real numbers.
• They have smooth, continuous curves.
• The degree of the polynomial tells you the maximum number of roots or zeros it may have.

Polynomials are used extensively in science and engineering to approximate values and solve problems.

Rational Functions

Rational functions are the quotient of two polynomials. They are typically expressed as \( \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) eq 0 \). An example is \( f(x) = \frac{x^2 - 1}{x + 2} \).

Rational functions can be more complex than polynomial functions, often containing asymptotes (lines that the graph approaches but never touches) and holes (undefined points). Here are some main features:

• Their domain excludes values that make the denominator zero.
• Their behavior can change dramatically around these undefined points.
• They help in modeling ratios and rates of change, like speed.

Rational functions are useful in analyzing mathematical limits and behavior near undefined points in calculus.

Piecewise Linear Functions

Piecewise linear functions are defined by different linear equations over specified intervals. They have multiple sections, each described by a linear function for a particular range of the input variable. A simple piecewise function like \( f(x) \) could be defined as:

• \( f(x) = x + 2 \) for \( x < 0 \)
• \( f(x) = -x + 5 \) for \( x \geq 0 \)

These functions are favored in the real world for their simplicity and straightforward nature in modeling situations where a total range is divided into segments.

Some advantages include:

• They can easily represent changes in a scenario over time.
• Their graphs consist of straight line segments which are easy to interpret.
• They are computationally less intensive to evaluate within each interval.

In practical terms, they often model real-life situations like tax brackets, shipping rates, or cost evaluations with stepped pricing systems.