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

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.

A polynomial is an algebraic function that can be written as Each a k 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.

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 (-∞, ∞).

The range (set of outputs produced) of a polynomial varies with its degree. Degree = 0 Degree = nonzero, even Degree = odd

Degree = 0  range = {#} Degree = even (≠ 0)  range = (-∞, max] or [min, ∞) Degree = odd  range = (-∞, ∞)

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.

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.

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.

Domain of : n even  {x: g(x) ≥ 0 and g(x) is defined} n odd  {x: g(x) is defined}

A function of the form where b > 0 is an exponential function. If b 1, the function is increasing.

The domain is all real numbers: (-∞, ∞). The range is all positive numbers: (0, ∞). The point (0, 1) is on every b x curve. The natural exponential function is e x. This function has many nice calculus properties.

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.

Domain = all positive numbers = (0, ∞) Range = all real numbers = (-∞, ∞)

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)

The six trigonometric functions of interest in our Calculus class are: We focus on sine and cosine.

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.

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

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.

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