
For real let , then
A) f is one-one and onto R
B) f is neither one-one nor onto R
C) f is one-one but not onto R
D) f is onto R but not one-one
Answer
511.8k+ views
Hint:The function is one-one if the function is strictly increasing. Differentiate the function and check for increasing and decreasing. A function is onto when for every element in the codomain there is at least one element in the domain.
Complete step-by-step answer:
We are given a function
Here function is defined from R to R that is
We have to check whether the function is one-one and onto or not.
First, we check for onto.
Let
Now, we see that we have a cubic polynomial in . We know that a polynomial of odd degree has at least one real root to any that belongs to codomain, then some which belong to the domain such that
Hence, function is onto.
Now, we check for one-one.
We know that if the function is strictly increasing then it is a one-one function.
We have
If then function is strictly increasing.
Differentiate the function with respect to .
Use
Since, square of any number is always positive therefore,
It means the function is increasing.
Therefore, is one-one.
Hence, is one-one and onto R.
Option (A) is correct.
Note:We can also use the horizontal line test to check if the function is one-one or not.
The horizontal line test states that if a line is drawn parallel to the x-axis and it cuts the graph exactly at one point then the function is one-one. If the line cuts exactly at one point it means for every y-value in the function, there is a unique x-value.
Complete step-by-step answer:
We are given a function
Here function is defined from R to R that is
We have to check whether the function is one-one and onto or not.
First, we check for onto.
Let
Now, we see that we have a cubic polynomial in
Hence, function is onto.
Now, we check for one-one.
We know that if the function is strictly increasing then it is a one-one function.
We have
If
Differentiate the function with respect to
Use
Since, square of any number is always positive therefore,
It means the function is increasing.
Therefore,
Hence,
Option (A) is correct.
Note:We can also use the horizontal line test to check if the function is one-one or not.
The horizontal line test states that if a line is drawn parallel to the x-axis and it cuts the graph exactly at one point then the function is one-one. If the line cuts exactly at one point it means for every y-value in the function, there is a unique x-value.
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