# Functions, Limits, and Continuity

### Contents

#### Terms

Composition  -  Given two functions f : ST and g : TU, the function gof : SU defined by applying f and then g is called the composition of g and f.
Continuous  -  A function f is said to be continuous at a pointx0 of its domain if it has a limit there, and that limit agrees with the value f (x0). In mathematical notation:

 f (x) = f (x0)

If the function is continuous at every point of its domain, then one simply says it is continuous.
Domain  -  The set of values that a function f takes as its input.
Elementary Functions  -  A standard library of functions including the linear, polynomial, rational, power, and trigonometric functions.
Function  -  A rule f that assigns to each element in a set S a unique element in a set T, which is written f : ST.
Image  -  The value f (x) which a function f assigns to a particular value x in its domain.
Interval  -  A set consisting of the real numbers between two fixed points, possibly including one or both of these endpoints. An open interval(a, b) is the set of real numbers x such that a < x < b (excluding the endpoints). A closed interval[a, b] is the set of real numbers x such that axb (including the endpoints).
Invertible  -  A term that describes a function f : ST such that there exists a function g : TS with (gof )(x) = x for each element xâààS.
Limit  -  The value that a function f (x) approaches as x approaches a particular value x0. This is the intuition behind a more rigorous definition.
Linear Function  -  A function of the form f (x) = ax + b, where a and b are real numbers. The graph of such a function is a straight line.
Polynomial Function  -  A function of the form f (x) = anxn + ... + a1x + a0 for real numbers a0,…an.
Power Function  -  A function of the form f (t) = Crt that is used to model exponential growth or decay.
Logarithm  -  The inverse of a power function f (t) = Crt is the logarithm with base r, denoted logr(t).
Range  -  The set within which the output of a function f lies.
Rational Function  -  A function that is formed by taking the quotient of two polynomials.
Set  -  A collection of objects (which are called elements).
Slope  -  The number a for the linear function f (x) = ax + b, indicating the steepness of the graph of f.
Trigonometric Function  -  A periodic function involving sines, cosines, tangents, or their reciprocals or inverses.
y-intercept  -  The number b = f (0) for a linear function f (x) = ax + b, indicating the vertical coordinate of the intersection point of the graph of f with the y-axis.