1.1 Four Ways To Represent A Function

Front 1

Function

Back 1

A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E.

Hint 1

f(x)

Front 2

Domain

Back 2

The set D of the function

Front 3

f(x)

Back 3

The value of f at x and is read "f of x"

Front 4

Independent variable

Back 4

A symbol that represents an arbitrary number in the domain of a function f

Front 5

Dependent variable

Back 5

A symbol that represents a number in the range of f

Front 6

Machine

Back 6

Helpful way to think of functions.

If x is in the domain of the function f, then when x enters the machine, it's accepted as an input and the machine produces an output f(x) according to the rule of the function.

Thus the domain is the set of all possible inputs and the range is the set of all possible outputs.

Front 7

Arrow diagram

Back 7

Another helpful way to look at functions.

Each arrow connects an element of D to an element of E. The arrow indicates that f(x) is associated with x, f(a) is associated with a, and so on.

Front 8

Graph

Back 8

If f is a function with domain D, then its graph is the set of ordered pairs:

{(x,f(x)) | x is in D}

In other words, the graph of f consists of all points (x,y) in the coordinate plane such that y=f(x) and x is in the domain of f.

Front 9

Difference quotient

Back 9

The average rate of change of f(x) between x=a and x=a+h.

Front 10

The Vertical Line Test

Back 10

A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.

Front 11

Piecewise defined functions

Back 11

Functions defined by different formulas in different parts of their domains.

You would follow the rules to find the coordinates necessary to graph the function.

Hint 11

Front 12

Absolute value

Back 12

The distance from a to 0. Denoted by |a|. Always positive or 0.

Hint 12

Front 13

Step functions

Back 13

Piecewise defined functions that jump from one value to the next.

Front 14

Even function

Back 14

When f(-x)=f(x) for every number x in its domain

In other words, it is a mirror image in respect to the y-axis. Usually parabolas.

Front 15

Odd function

Back 15

When f(-x)=-f(x) for every number x in its domain

In other words, one half of it is the mirror image of the other half turned 180° about the origin. Symmetric about the origin.

Front 16

Periodic function

Back 16

When a function has a positive constant T such that f(x+T)=f(x) for all values of x in the domain of f.

Front 17

Period

Back 17

The smallest value for which T is true in a periodic function.

Front 18

Increasing

Back 18

A function f is increasing on an interval I if f(x1)<f(x2).

Would be denoted by [ or ( and the x-coordinate where the decreasing begins, comma, the x-coordinate where the decreasing stops, and ) or ].

Front 19

Decreasing

Back 19

A function f is decreasing on an interval I if f(x1)>f(x2).