1. Calculus, Section 1.3

2. The Basic Classes of Functions
It is impossible to describe all possible functions. However, calculus makes no attempt to deal with all functions. The techniques of calculus, powerful and general as they are, apply only to functions that are sufficiently well-behaved*. Fortunately, such functions are adequate for a vast range of applications.
*We shall see what “well-behaved” means when we study the derivative in Chapter 3.

3. The Basic Classes of Functions
Most of the functions considered in this text are constructed from the following familiar classes of well-behaved functions:
Polynomials
Rational Functions
Algebraic Functions
Exponential Functions
Logarithmic Functions
Trigonometric Functions
Inverse Trigonometric Functions
We shall refer to these as the basic functions.

4. Polynomials
For any real number 𝑚, 𝑓 𝑥 = 𝑥 𝑚 is called the power function with exponent 𝑚. A polynomial is a sum of multiples of power functions with whole-number exponents. Two examples are the following: 𝑓 𝑥 = 𝑥 5 −5 𝑥 3 +4𝑥 𝑔 𝑡 =7 𝑡 6 + 𝑡 3 −3𝑡−1

5. Polynomials The general polynomial in the variable 𝑥 may be written
𝑃 𝑥 = 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +⋯+ 𝑎 1 𝑥+ 𝑎 0
The numbers 𝑎 𝑛 , 𝑎 𝑛−1 , , 𝑎 1 , 𝑎 0 are called coefficients.
The degree of 𝑃(𝑥) is 𝑛. (This assumes that 𝑎 𝑛 ≠0.)
The coefficient 𝑎 𝑛 is called the leading coefficient.
The domain of 𝑃(𝑥) is ℝ.
Recall from Section 1.1 that the symbol ℝ represents the set of real numbers.

6. Rational Functions
A rational function is a quotient of two polynomials: 𝑓 𝑥 = 𝑃(𝑥) 𝑄(𝑥) , where 𝑃(𝑥) and 𝑄(𝑥) are polynomials. The domain of 𝑓(𝑥) is the set of all real numbers 𝑥 such that 𝑄(𝑥)≠0.

7. Rational Functions
Some examples of the rational functions include the following: 𝑓 𝑥 = 1 𝑥 2 , which has domain 𝑥∈ℝ:𝑥≠0 ℎ 𝑡 = 7 𝑡 6 + 𝑡 3 −3𝑡−1 𝑡 2 −1 , which has domain 𝑡∈ℝ: 𝑡 2 −1≠0 = 𝑡∈ℝ:𝑡≠±1 Notice that, because the basic definition of a rational function is a quotient of two polynomials 𝑃(𝑥) 𝑄(𝑥) , every polynomial is also a rational function since 𝑄 𝑥 =1 is a polynomial.

8. Algebraic Functions
An algebraic function is produced by taking sums, products, and quotients of roots of polynomials and rational functions. A number x belongs to the domain of an algebraic function if each term in the formula is defined and the result does not involve division by zero.

9. Algebraic Functions
Some examples of algebraic functions are the following: 𝑓 𝑥 = 1−3 𝑥 2 − 𝑥 4 , where has domain 𝑥∈ℝ:1−3 𝑥 2 − 𝑥 4 ≥0 𝑔 𝑡 = 𝑡 −2 −2 , which has domain 𝑡∈ℝ:𝑡≥0∧𝑡≠4 ℎ 𝑧 = 𝑧+ 𝑧 −5/3 5 𝑧 3 − 𝑧 , which has domain 𝑧∈ℝ:𝑧>0∧5 𝑧 3 − 𝑧 ≠0 . The ∧ symbol in the domains of 𝑔 and ℎ above stands for the logical “and”. Alternatively, the ∨ symbol stands for the logical “or”.

10. Algebraic Functions*
In a more general sense, algebraic functions are defined by polynomial equations between 𝑥 and 𝑦. In this case, we say that 𝑦 is implicitly defined as a function of 𝑥.
For example, the equation 𝑦 4 +2 𝑥 2 𝑦+ 𝑥 4 =1 defines 𝑦 implicitly as a function of 𝑥.
Implicit functions will be discussed further in Sections 3.10 and 3.11, where we will discuss implicit differentiation and related rates. These topics are highly applicable in analyzing real-world phenomena.
*A function that is not algebraic is called a transcendental function.

11. Exponential Functions
The function 𝑓 𝑥 = 𝑏 𝑥 , where 𝑏>0, is called the exponential function with base 𝑏. Some examples are the following:
𝑓 𝑥 = 2 𝑥 𝑔 𝑡 = 10 𝑡 ℎ 𝑥 = 𝑥 𝑝 𝑡 = 𝑡
All exponential functions have domain ℝ. Recall that the natural exponential function is 𝑓 𝑥 = 𝑒 𝑥 , where 𝑒≈ The number 𝑒 is commonly called Euler’s number.* Exponential functions are explained in greater detail in Section 1.6.
*In calculus, the importance of 𝑒 becomes increasingly obvious, as the natural exponential function is its own derivative.

12. Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. The topic of inverse functions is reviewed in Section 1.5, and logarithmic functions are explained in greater detail in Section 1.6, alongside exponential functions.

13. Trigonometric Functions and Inverse Trigonometric Functions
Trigonometric functions are functions built from sin 𝑥 and cos 𝑥 . Trigonometric functions are discussed in Section 1.4, and inverse trigonometric functions are discussed in Section 1.5.

14. Constructing New Functions
Given functions 𝑓 and 𝑔, we can construct new functions by forming the sum, difference, product, and quotient functions: 𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔(𝑥) 𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔(𝑥) 𝑓𝑔 𝑥 =𝑓 𝑥 𝑔(𝑥) 𝑓 𝑔 𝑥 = 𝑓(𝑥) 𝑔(𝑥) , 𝑔(𝑥)≠0

15. Linear Combinations
We can also multiply functions by constants. A function of the form 𝑐 1 𝑓 𝑥 + 𝑐 2 𝑔(𝑥), where 𝑐 1 and 𝑐 2 are constants, is called a linear combination of 𝑓(𝑥) and 𝑔(𝑥).

16. In-Class Assignment 1.3.1
Given 𝑓 𝑥 = 1 𝑥 and 𝑔 𝑥 = 𝑥 −4 , calculate the following: 𝑓+𝑔 𝑥 𝑓−𝑔 𝑥 𝑓𝑔 𝑥 𝑓 𝑔 𝑥

17. Composition
The composition of 𝑓 and 𝑔 is the function 𝑓∘𝑔 defined by 𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥 . The domain of 𝑓∘𝑔 is 𝑥∈ℝ:𝑥∈𝑑𝑜𝑚(𝑔)∧𝑔(𝑥)∈𝑑𝑜𝑚(𝑓)

18. In-Class Assignment 1.3.2
Given 𝑓 𝑥 = 1 𝑥 2 +1 and 𝑔 𝑥 = 𝑥 −2 , calculate the following: 𝑓∘𝑔 𝑥 𝑔∘𝑓 𝑥

19. Elementary Functions
Whenever a function is constructed by applying the operations of addition, subtraction, multiplication, division, and composition of basic functions, it is said to be an elementary function. The following functions are elementary: 𝑓 𝑥 = 2𝑥+ sin 𝑥 𝑓 𝑥 = 10 𝑥 𝑓 𝑥 = 1+ 𝑥 −1 1+ cos 𝑥