College Algebra Definitions Chapter 2

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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.

Function Notation

Function Generic Venn Diagram

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.

Zero of a Function

A *zero of a function* f is a domain element k for which
f(k) = 0.

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).

Even Function

A function f is an *even function* if,
f(x) = f(-x)
for all domain elements x

Odd Function

A function f is an *odd function* if, f(x) = -f(-x) for all domain elements x.

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.

Inverse of a function

Let f be a function with domain A and range B. Then the inverse of the function, if it exists, is a function named f⁻¹, with domain B and range A with the property shown above.

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