different types of functions
DESCRIPTION
by my teacher Ms. ShelamarTRANSCRIPT
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Different Types
of Functions
prepared by: Shielamar L. Labiscase
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The set of real numbers
SET NOTATION
A set is collection of objects.
The objects in a set are elements or
members of the set.
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ROSTER METHOD
of writing a set encloses the elements of the set in braces, {}.
Ex. The set of natural numbers less than 11.
A={1,2,3,4,5,6,7,8,9,10}
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A={1,2,3,4,5,6,7,8,9,10}This set has limited number of
elements and is an example of finite set.
To express the fact that 10 is an element of the set, use the symbol , i.e.
10 A.
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A second way to denote a set is use to use set-builder notation, where the set is written as
A = { x/x is a natural number less than 11}
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Domain of the function
The domain of the function is the set
of consisting of all values of x.
Determine the domain of f(x) =
3/(x+7)
Therefore, the dom(f) is the set of all
real numbers except -7.{x/x=/ -7}
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Range of the function
• The range of the functions is the set consisting of all the second components in each element of the set.
–Find the of f(x) = 3/ x+7
• Range (f) = {y/y=/0}
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Types of Functions
Constant FunctionIdentity FunctionPolynomial FunctionAbsolute Value
FunctionSquare Root FunctionRational FunctionGreatest Integer
FunctionPiece-wise FunctionUnit Step Function
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Constant Functions
The constant function C is function with the range of which is consist of a single number k for all real numbers x in its domain. In symbol C(x) = k.
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Graph f(x) = 5 x x -2 -1 0 1 2
y f(x) 5 5 5 5 5
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What is the constant function of this graph?
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Domain and Range of a Function
• The domain of the constant function is all real numbers
• The range is the constant k. In this function is equal to 5
• The graph is a horizontal line.
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Identity Functions
The identity function I is defined by I(x) = x.
The domain is the set of real numbers.
The range of the identity function is also the set of all real numbers.
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Graph f(x) = x
X -2 -1 0 1 2y -2 -1 0 1 2
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Polynomial Functions
A polynomial in the variable x is a function that can be written in the form,
where an, an-1 , ..., a2, a1, a0 are constants. We call the term containing the highest power of x(i.e. anxn) the leading term, and we call an the leading coefficient. The degree of the polynomial is the power of x in the leading term.
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Name of the functionDegree of the
PolynomialName of the
function
0 Constant function
1 Linear function
2 Quadratic function
3 Cubic Function
4 Quartic Function
5 Quintic Function
n (where n > 5) n (where n > 5)
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Linear Function
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Domain and Range of Linear Function
The domain of a linear function is the set of real numbers {x/x is a real number}.
The range of the linear function is the set of real numbers {y/y is a real number}.
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Quadratic function: f(x) = x^2
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Domain and Range of Quadratic Function
Domain : {x\x is a real number} Range: If a > 0, {f(x)/f(x) ≥ k}
If a < 0, {f(x)/f(x) ≤ k}
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Cubic
Function
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Absolute Value Function
It is defined by f(x) = /x/
• The domain of the absolute value function is all real numbers.
• The range is all non negative numbers
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Graph of f(x) = /x/Domain: {x/x is a real number}Range: {f(x)/f(x) ≥ 0}
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Graph in one Cartesian Coordinate Plane
• y =
• y =
• y = /x/ + 2
• y = /x/ - 2
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Graph of y= • To graph y = , simply shift the graph
of y = /x/, A units to the left.• To graph y = , simply shift the graph
of y = /x/, A units to the right.
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Graph of y= • To graph y = , simply shift
the graph of y = /x/, B units upward.
• To graph y = , simply shift the graph of y = /x/, B units downward.
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Domain: {x/x is a real number}Range: {y/y ≥ 1}
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Absolute value function
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Domain: {x/x is a real number}Range: {y/y ≤ -2}
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Square Root Functions
xxf )(
Domain :Range:Graph:
0xx
0yy
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Graph and determine the
domain and range
• h(x) =
• H(x) =
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Rational Functions
xxf
1)(
Domain: (x)/x ≠
0}
Range:
{f(x)/f(x) ≠ 0}
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Definition of AsymptoteAn asymptote is an imaginary line being approached but never touched or intersected by a graph as it goes through infinity
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What is the domain and range of the rational function?
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What is the domain and range of the rational
function?
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Greatest Integer Function: graph
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Domain and range of Greatest Integer
FunctionThe domain of G(x) = [x] is the
set of real numbers.
The range is the set of
integers.
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Piece-wise function
• when x is less than 2, it gives x2,• when x is exactly 2 it gives 6• when x is more than 2 and less than or equal
to 6 it gives the line 10-x
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It looks like this:
• a solid dot means "including",an open dot means "not including")
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Domain and Range of the Piece-wise Function
• The Domain (all the values that can go into the function) is all Real Numbers up to and including 6, which we can write like this
• Dom(f) = (-∞, 6] (using Interval Notation)• Dom(f) = {x | x ≤ 6} (using
Set Builder Notation)
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Signum Function
The sign of a real number, also called sgn or signum, is for a negative number (i.e., one with a minus sign " "), 0 for the number zero, or for apositive number (i.e., one with a plus sign " ").
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In other words, for real ,
• For real, this can be written
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Domain and range of a
Signum function
Domain : {x/x is a set of real
number}
Range : {-1, 0, 1}
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Unit Step Function
• It is defined as
• Domain: {x/x is the set of real numbers}
• Range: {0,1}
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Thank you for listening.
God bless you all.
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