This section is a field guide to all of the functions with which we must be proficient in a Calculus course.

Section 1.3The Functions of

Calculus

Definitions

An elementary function is one built from certain basic elements using certain allowed operations.

An algebraic function is one using only the operations of +, -, *, /, and powers or radicals.

Transcendental functions include exponential, logarithmic, and trigonometric functions.

Algebraic Functions:

Polynomials

A polynomial is an algebraic function that can be written as

Each ak is called a coefficient, and can have any real number value.

The degree of a polynomial is the largest exponent for which the coefficient is not 0.

Polynomials Continued

Polynomial graphs are “smooth” everywhere – they have no “sharp points”.

Polynomial graphs have no “breaks” in them – they are continuous everywhere.

The domain (set of valid input values) of a polynomial is all real numbers, which is (-∞, ∞).

Polynomials Continued

The range (set of outputs produced) of a polynomial varies with its degree.

Degree = 0 Degree = nonzero, even Degree = odd

Polynomials Continued

Degree = 0 range = {#}

Degree = even (≠ 0) range = (-∞, max] or [min, ∞)

Degree = odd range = (-∞, ∞)

Algebraic Functions:

Rational Functions

A rational function is a function that can be written as:

Here, both p and q are polynomials.

The domain is ; the range varies a lot from function to function.

Rational Functions Continued

For any x value for which q(x) = 0 but p(x) ≠ 0, the rational function f has a vertical asymptote.

As means x = c is a V.A.

Algebraic Functions:

Radical Functions

A function of the form is called a radical function.

The “inside” function, g(x), is called the radicand.

The index of the radical is n. For a square-root function, the index is 2 even though it is not written in the radical notation.

Radical Functions Continued

Domain of :

n even {x: g(x) ≥ 0 and g(x) is defined}

n odd {x: g(x) is defined}

Transcendental Functions:

Exponential Functions

A function of the form where b > 0 is an exponential function.

If b <1, the function is decreasing. If b > 1, the function is increasing.

Exponential FunctionsContinued

The domain is all real numbers: (-∞, ∞).

The range is all positive numbers: (0, ∞).

The point (0, 1) is on every bx curve.

The natural exponential function is ex. This function has many nice calculus properties.

Transcendental Functions:

Logarithms

The logarithm function with base b (where b > 0),

is the inverse of the exponential function defined by .

This means

If b > 1, the function increases and if b < 1 the function decreases.

Logarithm Functions:Continued

Domain = all positive numbers = (0, ∞)

Range = all real numbers = (-∞, ∞)

Logarithm Functions:Continued

The base 10 logarithm is called the common logarithm and is denoted as log(x).

The base e logarithm is called the natural logarithm and is denoted as ln(x).

All logarithm functions pass through the point (1, 0)

Transcendental Functions:

Trig. Functions

The six trigonometric functions of interest in our Calculus class are:

We focus on sine and cosine.

Trigonometric FunctionsContinued

f(t) = sin(t) and g(t) = cos(t) are defined in terms of the arc length t (measured in radians), and the corresponding point on the unit circle.

http://cerebro.cs.xu.edu/~staat/Handouts/UnitCircle.pdf

Trigonometric FunctionsContinued

Plotting these “special angles”, we get the following graphs:

Trigonometric FunctionsContinued

Domain of sine and cosine is (-∞, ∞).

The range of sine and cosine is [-1, 1].

The other functions are all defined in terms of sine and cosine, so knowing these two well allows us to work with any of the others.

Trigonometric FunctionsContinued

The domain of each of these is determined by the fact that denominators cannot be 0 and the following facts: