1. Let A = { 2, 3, 4};
Let B = {4, 9,16};
F: x->y, f(x) = x^2;
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How did you determine that it is indeed a function?
To check if it is a function, each element of set A must be related to one and only one
element of set B. This is the case for our function
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Share your opinion on whether it is one-one and onto explain, why?
It is a one to one function because every element of A is mapped to a unique element
of B. That is for each element in the domain, there is only one element in the range. It
is also an onto function because the range of the function is equal to its codomain.
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Does it have an inverse? Justify your choice with the above representation.
Since it is bijective, it must have an inverse, that is, f^-1 : B->A is defined by f^-1 (y)
= sqrt(y).
2. A function is a relationship between a set of inputs and a set of potential outputs,
characterized by the fact that each input corresponds to precisely one output. This
means that for any given input, there is only one output. For instance, the function f(x)
= x + 2 illustrates this concept. For every input value of x, there is exactly one output
value, which is the value of x + 2. A counterexample of a function would be a relation
that assigns multiple outputs to a single input, thus violating the property that each input
is related to exactly one output.
Injective Function
An injective function, also known as a one-to-one function, is a function in which each element
of the domain maps to a distinct element in the codomain. Put simply, no two different elements
in the domain can map to the same element in the codomain.
f(x) = x^2 is not an example of an injective function since different elements such as -1 and 1
both map to the same output value of 1 when squared. An example of an injective function
would be f(x) = 2x, where each unique input value maps to a unique output value. For each
unique input value of x, there is a corresponding unique output value.
Surjective Function
A surjective function, also known as an onto function, is a function in which every element of
the codomain is mapped to by at least one element from the domain. This means that for every
element in the codomain, there exists at least one pre-image in the domain.
For instance, the function f(x) = e^x is onto because every real number y can be written as e^x
for some real number x.
However, not all functions are onto. A counterexample is the function f(x) = x^2 where x
belongs to the set of real numbers (ℝ). The range of this function excludes negative numbers;
therefore, it is not surjective since not every element of the codomain has a pre-image in the
domain.
Bijective Function
A bijective function is a type of function that is both injective and surjective. This implies that
each element in the domain is mapped to a unique element in the codomain, and every element
in the codomain has precisely one pre-image in the domain.
For instance, the function f(x) = x + 5 is bijective because it fulfills both the criteria for
injectivity and surjectivity.
Conversely, the function f(x) = x^2 is not bijective. It is not injective since it maps both x and
-x to the same value in the codomain.