can a constant function be one - one or onto if so when
Answers
Answer:
Step-by-step explanation:
A constant function y = C, C constant can NEVER be one-to-one since for different values of x we have the same value of y, namely C.
On the other hand every constant function y = C IS onto since för Each value of y, in this case y = C, we can find at least one x such that f(x) = C.
To Find:
Whether a constant function is one-one or onto. If yes, then find the condition for being one-one or onto.
Solution:
Let us assume that the domain and codomain of the constant function be the set of all real numbers.
A constant function is one where the output is always the same, regardless of the value of the input i.e., .
Let, for example, a constant function be .
One-one function:
A function is said to be one-one if every element of the range of
corresponds to exactly one element of the domain of
.
Thus, will be one-one if every
value that can come out of this function (i.e., range of the function) comes from only one
value.
Here, the range is just while
can be any real number. Hence, the
value that comes out of this function will always be
for any
value.
Therefore, this function is not one-one.
However, if we change the domain to include only a single value, say , then the function
will take 1 as an input and produce 5 as an output. In that case, it would be one-one.
Onto function:
A function is said to be onto if every element of the codomain of
corresponds to at least one element of the domain of
.
Thus, will be onto if every
value is an image of at least one
value. Since we have taken the codomain as the set of real numbers, all the
values except 5, will not be mapped with the
values.
Therefore, this function is not one-one.
However, if we change the codomain to be just , then 5 will be the image of all
values. In that case, the function would be onto.
Thus, a constant function is neither one-one nor onto unless its domain and codomain are restricted to a single value.
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