This document defines and describes different types of functions. It states that a function relates inputs to exactly one output and has a domain and codomain or range. The main types of functions discussed are: one-to-one functions where each input maps to a unique output; many-to-one functions where multiple inputs can map to the same output; onto functions where each element in the codomain has a pre-image in the domain; identity functions where the output equals the input; and constant and quadratic functions defined by polynomial equations. Cubic functions are also polynomial functions of degree three.

1. FUNCTIONS
B Y :
F . J H A N S I R A N I

2. DEFINITIONOF FUNCTIONS
• A function is a relationship between inputs where each input is related to
exactly one output.
• Every function has a domain and codomain or range.
• A function is generally denoted by f(x) where x is the input. The general
representation of a function is y = f(x).
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3. TYPES OF FUNCTIONS
 One-one Function (Injective Function)
 Many-one Function
 Onto Function (Surjective Function)
 Into Function
 Identity Function
 Constant Function
 Quadratic Function
 Cubic Function

4. ONE-ONE FUNCTION
One-one Function (Injective Function).
If each element in the domain of a function has a
distinct image in the co-domain, the function is
said to be a one-one function.
 For example ,f(x) = 3x + 5, is one-one.
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5. MANY-ONE FUNCTION
If there are at least two elements in the domain whose images
are the same, the function is known as many to one.
For example, f : R given by f(x) = x2 + 1, is many-one.
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6. ONTO- FUNCTION
 Onto Function (Surjective Function)
 A function is called an onto function if each element in
the co-domain has at least one pre-image in the domain.
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7. INTO- FUNCTION
• If there exists at least one element in the co-domain, which is not an image of any
element in the domain, then the function will be into function.
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8. IDENTITY FUNCTION
Let R be the set of real numbers. If the
function f: R→R is defined as f(x) = y = x, for
x ∈ R, then the function is known as Identity
function.
 The domain and the range being R. The graph
is always a straight line and passes through the
origin.
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9. CONSTANT FUNCTION
If the function f: R→R is defined as f(x) = y = c, for x ∈ R
and c is a constant in R, then such function is known as
Constant function.
The domain of the function f is R and its range is a
constant, c. Plotting a graph, we find a straight line
parallel to the x-axis.
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10. QUADRATICFUNCTION
• If the degree of the polynomial function is two, then it is
a quadratic function. It is expressed as f(x) = ax2 + bx + c,
where a ≠ 0 and a, b, c are constant & x is a variable.
• The domain and the range are R. The graphical
representation of a quadratic function say, f(x) = x2 – 4 is
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11. CUBICFUNCTION:
 A cubic polynomial function is a polynomial of degree three
and can be denoted by f(x) = ax3 + bx2 + cx +d, where a ≠ 0
and a, b, c, and d are constant & x is a variable.
 Graph for f(x) = y = x3 – 5. The domain and the range are R.
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12. THANK YOU
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