Onto function means Range of the function should belong to the co- domain of the function, which is a key point of the onto function.Logarithmic vs Exponential | Exponential Function vs Logarithmic Functionįunctions are one of the most important classes of mathematical objects, which are extensively used in almost all subfields of mathematics.
We can verify whether any function is one-one by finding derivation of the function on $f\left( x \right)$ as well, If $f\left( x \right)>0$ only or $f'\left( x \right)<0$ only for the given domain then $f\left( x \right)$ is one-one, otherwise $f\left( x \right)$ will not be one-one. So, don’t confuse with the examples given in the problem, there are other examples as well. So, it can be another way as well.There are infinite examples of the functions asked in the problem. Note: Another approach to get any function to be one, we can use graphical approach, means draw the curve $y=f\left( x \right)$ as supposed in all the four option, if a line parallel to x cutting the graph at two or more points, then it will not be one-one. So, the function $f:N\to N$, given by $f\left( 1 \right)=f\left( 2 \right)=1$ is not one-one but onto. Here, y is a natural number for every ‘y’, there is a value of x which is a natural number.
Let $f\left( x \right)=y$, such that $y\in N$. Since, different elements 1 and 2 have the same image ‘1’. (iv) Let the function $f:N\to N$, given by $f\left( 1 \right)=f\left( 2 \right)=1$ Mathematically, one-one is given for any function $f\left( x \right)$ as if $f\left( $ is not one-one and not onto as well for $x:R\to R$. mapping of elements of range and domain are unique. Hint:One-one function means every domain has distinct range i.e.
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