Representing Functions in Calculus: Types, Properties, and Transformations

Section 1.2 — Representing Functions

Introduction

This section introduces the foundational concept of functions in calculus, focusing on their types, properties, and graphical representations. Understanding these basic function types and their transformations is essential for further study in calculus.

Algebraic Functions

Types of Algebraic Functions

Algebraic functions are functions constructed using algebraic operations (addition, subtraction, multiplication, division, and taking roots) on variables and constants. The main types include:

• Polynomial Functions: Functions of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $n$ is a nonnegative integer and $a_0, a_1, \ldots, a_n$ are constants called coefficients.

• Rational Functions: Functions that are ratios of two polynomials, e.g., $f(x) = \frac{x+1}{x-2}$.

• Power Functions: Functions of the form $f(x) = x^r$, where $r$ is any real number (including roots, such as $f(x) = \sqrt{x}$ or $x^{1/2}$).

Polynomial Functions

• Definition: A polynomial of degree n is a function $P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$.

• Degree: The highest power of $x$ with a nonzero coefficient.

• Coefficients: The constants $a_0, a_1, \ldots, a_n$.

• Examples:

  • Cubic: $y = x^3 - x + 1$

  • Quartic: $y = x^4 - 3x^2 + x$

  • Quartic: $y = 3x^4 - 2.5x^3$

Root Functions

Even Root Functions

Even root functions involve taking an even root (such as square root, fourth root) of $x$. Their domain is restricted to nonnegative values of $x$.

• Examples:

  • $y = x^{1/2} = \sqrt{x}$

  • $y = x^{1/4} = \sqrt[4]{x}$

  • $y = x^{1/6} = \sqrt[6]{x}$

• Domain: $x \geq 0$

• Graph: All even root functions pass through $(0,0)$ and $(1,1)$, but increase more slowly as the root increases.

Odd Root Functions

Odd root functions involve taking an odd root (such as cube root, fifth root) of $x$. Their domain includes all real numbers.

• Examples:

  • $y = x^{1/3} = \sqrt[3]{x}$

  • $y = x^{1/5} = \sqrt[5]{x}$

• Domain: $x \in \mathbb{R}$ (all real numbers)

• Graph: Odd root functions are symmetric about the origin (odd functions) and pass through $(0,0)$ and $(1,1)$.

• Recall: $y = x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$

Summary Table: Types of Algebraic Functions

Type	Definition	Example
Polynomial	Sum of powers of $x$ with constant coefficients	$f(x) = x^4 - 3x$
Rational	Ratio of two polynomials	$f(x) = \frac{x+1}{x-2}$
Power	$x$ raised to any real exponent	$f(x) = \sqrt{x}$ or $x^{1/2}$

Additional info:

• These notes are based on Section 1.2 of a standard Calculus textbook (Early Transcendentals), focusing on the representation and classification of functions.

• Understanding the properties and graphs of these basic functions is crucial for later topics such as limits, derivatives, and integrals.