Question

can a constant function be one - one or onto if so when​

Answer 1

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.

Answer 2

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., $f(x)=k$.

Let, for example, a constant function be $f(x)=5$.

One-one function:

A function $g$ is said to be one-one if every element of the range of $g$ corresponds to exactly one element of the domain of $g$.

Thus, $f(x)=5$ will be one-one if every $y-$value that can come out of this function (i.e., range of the function) comes from only one $x-$value.

Here, the range is just $\[\left\{ 5 \right\}\]$ while $x$ can be any real number. Hence, the $y-$value that comes out of this function will always be $5$ for any $x-$value.

Therefore, this function is not one-one.

However, if we change the domain to include only a single value, say $\[\left\{ 1 \right\}\]$,    then the function $f(x)=5$ will take 1 as an input and produce 5 as an output. In that case, it would be one-one.

Onto function:

A function $g$ is said to be onto if every element of the codomain of $g$ corresponds to at least one element of the domain of $g$.

Thus,  $f(x)=5$ will be onto if every  $y-$value is an image of at least one $x-$value. Since we have taken the codomain as the set of real numbers, all the $y-$values except 5, will not be mapped with the $x-$values.

Therefore, this function is not one-one.

However, if we change the codomain to be just $\[\left\{ 5 \right\}\]$, then 5 will be the image of all $x-$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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