2. Important Information
• Functions are mathematical ideas that take one or more variables and produce a variable.
• In an abstract mathematical sense, a function is a mapping of some domain onto some range.
• For each item in the domain, there is a corresponding item in the range of the function.
• Thus the domain is all of the possible inputs to the function and the range is all of the possible outputs.
• Each item in the domain corresponds to a specific item in the range.
• However, an item in the range may correspond to multiple items in the domain.
3. List of Functions
1. squaring function f(x) = x2
2. cubing function f(x) = x3
3. absolute value function f(x) = |x|
4. square root function f(x) = sqrt(x)
5. cube root function f(x) = cubeRoot(x)
6. natural exponential function f(x) = ex
7. natural logarithmic function f(x) = ln(x)
4. Definition
• a function consists of an ordered triple of sets, written as (X,Y,F).
• X is the domain of the function
• Y is the codomain
• F is a set of ordered pairs
• A function basically gets an input and returns with an output like a machine
• The graph of a function is its set of ordered pairs.
• A relation is the set of ordered pairs
• A domain is the set of first coordinates of the ordered pairs (x values)
• A range is the set of second coordinates of the ordered pairs (y values)
8. History
The history of the function concept in mathematics is described by da Ponte (1992). The underlying
idea of a function dates back to the Persian mathematician, Sharaf al-Dīn al-Tūsī, in the 12th century.
In his analysis of the equation x3 + d = bx2 for example, he begins by changing the equation's form to
x2(b − x) = d. He then states that the question of whether the equation has a solution depends on
whether or not the “function” on the left side reaches the value d. To determine this, he finds a
maximum value for the function. Sharaf al-Din then states that if this value is less than d, there are no
positive solutions; if it is equal to d, then there is one solution; and if it is greater than d, then there are
two solutions.
