These definitions introduce the student to the concept of function.

### Function

A *function* consists of three things:

1) A set called the Domain

2) A set called the Range

3) A rule which associates each element of the domain with a unique element of the range.

### Satisfy the Rule

The coordinates of a point (a, b) are said to *satisfy the rule* of a function f if b = f(a).

### Graph of a function

The *graph of a function* is the set of all points whose coordinates satisfy the rule of the function.

### Graph of a Function (Preferred)

The *graph of a function* is the set of all points of the form

(a, f(a))

where a is an element of the domain and f(a) is the corresponding range element.

### x-intercept

An *x-intercept* of a graph in the Cartesian Coordinate System is a point where the graph intersects the x-axis.

### y-intercept

A *y-intercept* of a graph in the Cartesian Coordinate System is a point where the graph intersects the y-axis.

### Zero Function

The *zero function* z is the function defined by

z(x) = 0

for all x in the domain of z.

### Constant Function

A function f is called a *constant function* if its rule can be written as f(x) = k for some real number k.

### Linear Function

A *linear function* is a function whose rule may be written in the form f(x) = mx + b where m and b are real numbers.

### Identity Function

The *identity function* is the function I whose rule may be written in the form

I(x) = x.

### Squaring Function

The *squaring function* is the quadratic function f whose rule may be written in the form f(x) = x².

### Cubing Function

The *cubing function* is the function f whose rule may be written in the form

f(x) = x³.

### Quadratic Function

A *quadratic function* is a function whose rule may be written in the form

f(x) = ax² + bx + c

where a, b, and c are real numbers and a is not zero.

### Reciprocal Function

The *reciprocal function* is the function f whose rule may be written in the form shown above.

### Square Root Function

The *square root function* is the function sqrt whose rule may be written in the form

shown above.

### Absolute Value Function

The *absolute value function* is a function abs whose rule may be written in the form abs(x) = | x |.

### Exponential Base e Function

The *exponential base e function* is the function exp whose rule may be written in the form exp(x) = e×

where e is the irrational number approximately equal to 2.718281828...

### Logarithm Base e Function

The *logarithm base e function* is the function ln which is the inverse of the function exp.

### Piecewise Defined Function

A *piecewise defined function* is a function whose rule is different for different intervals of its domain.

### Increasing Function

A function f is *increasing on an interval* if, for any x₁and x₂ in the interval,

x₁< x₂ implies f(x₁) < f(x₂).

### Decreasing Function

A function f is *decreasing on an interval* if, for any x₁and x₂ in the interval,

x₁< x₂ implies f(x₁) > f(x₂).

### Sum of Functions

The *sum of two functions* f and g with the same domain is the function named (f+g) whose rule may be written as

(f+g)(x) = f(x) + g(x)

for all x in the common domain.

### Difference of Functions

The *difference of two functions* f and g with the same domain is the function named (f-g) whose rule may be written as

(f-g)(x) = f(x) - g(x)

for all x in the common domain.

### Product of functions

The *product of two functions* f and g with the same domain is the function named (fg) whose rule may be written as

(fg)(x) = [f(x)][g(x)]

for all x in the common domain.

### One-to-One Function

A function is called a *one-to-one function* if no element of the range is the associate of more than one domain element.

### Name for Composition of Two Functions

The composition of a function f with a function g is a function whose name is shown above.

### Composition of Two Functions

The composition of a function f with a function g is a function whose rule may be written in the form shown above.