Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$ f(x)=3^{x+1} $$

Step-by-Step Solution

Answer

The function \( f(x) = 3^{x+1} \) is neither a power function nor a polynomial function; it is an exponential function.

Step 1: Understand Power Function

A power function is in the form \( f(x) = ax^b \) where \( a \) and \( b \) are constants, and \( x \) is raised to a power \( b \). In this case, the variable \( x \) is an exponent of some constant.

Step 2: Understand Polynomial Function

A polynomial function is an expression that consists of variables and coefficients, composed using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables, such as \( f(x) = ax^n + bx^{n-1} + \, ... + cx + d \).

Step 3: Compare Given Function

Given the function \( f(x) = 3^{x+1} \), observe that the base 3 is raised to a power that involves the variable \( x \). This does not match the structure of a power function \( (ax^b) \) nor a polynomial function, as a polynomial involves variables raised to constant whole number exponents.

Step 4: Identify the Function Type

Since \( f(x) = 3^{x+1} \) does not match either the power function or polynomial function form, it must belong to another category. It is an exponential function, where the variable \( x \) is in the exponent.

Key Concepts

Power Functions

Power functions play a pivotal role in algebra and calculus. These functions follow the standard form \( f(x) = ax^b \), where:

• \( a \) is a constant, acting as the coefficient.
• \( b \) is also a constant, representing the exponent.
• \( x \) is the variable raised to the power \( b \).

For example, \( f(x) = 2x^3 \) qualifies as a power function, because it fits the pattern \( ax^b \) with \( a = 2 \) and \( b = 3 \). Power functions can model various real-world phenomena and are crucial in differentiating and integrating expressions in calculus. Understanding their structure helps in identifying them correctly.
Note, however, that if the variable is in the exponent position, as in \( 3^x \), it does not qualify as a power function.

Polynomial Functions

Polynomial functions are versatile and widely used in mathematics. Their standard equation is \( f(x) = ax^n + bx^{n-1} + \, ... + cx + d \), characterized by:

• Constants \( a, b, \ldots, d \) are coefficients.
• The powers of \( x \) are non-negative integers (e.g., 0, 1, 2, etc.).
• No variables appear in the exponent.

The simplest example is the linear function \( f(x) = x + 1 \), where the highest power of \( x \) is 1. Another example is a quadratic polynomial like \( f(x) = x^2 + 5x + 6 \).
Polynomial functions can be easily added, subtracted, and multiplied, making them useful in constructing complex mathematical models. They describe diverse concepts such as trajectories, growth patterns, and more. It's essential to recognize their form for accurate mathematical analysis.

Function Identification

Function identification involves recognizing the type of function based on its structure. Understanding the characteristics of power and polynomial functions helps greatly in this process. For instance, if a mathematical expression has a variable base raised to a constant power, it may align with either power or polynomial functions.
However, if the variable appears in the exponent, as in \( f(x) = 3^{x+1} \), it is not a power function nor a polynomial. Instead, this type of function is exponential since it has a constant base and a variable exponent. Recognizing this difference is crucial for solving problems correctly. Function identification helps in applying the correct mathematical rules and techniques for solving equations and understanding the nature of the graph, which is fundamental in calculus and algebra.