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For real x let , thenf(x)=x3+5x+1
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
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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 f(x)=x3+5x+1
Here function is defined from R to R that is f:RR
We have to check whether the function is one-one and onto or not.
First, we check for onto.
Let
f(x)=x3+5x+1=yf(x)=x3+5x+1y=0
Now, we see that we have a cubic polynomial in x. We know that a polynomial of odd degree has at least one real root to any y that belongs to codomain, then some x which belong to the domain such that f(x)=y
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 f(x)=x3+5x+1
If f(x)>0 then function is strictly increasing.
Differentiate the function with respect to x.
Use ddx(xn)=nxn1
f(x)=3x2+5
Since, square of any number is always positive therefore, f(x)>0
It means the function is increasing.
Therefore, f(x)is one-one.
Hence, f(x)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.
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