Question

For real x, let $$f(x)=x^3+5x+1$$, then
(a) f is onto R but not one-one
(b) f is one-one and onto R
(c) f is neither one-one nor onto R
(d) f is one-one but not onto R

Answer 1

Hint: We will use the definition of one-one function and onto function to solve this question. A function f is one-to-one if every element of the range of f corresponds to exactly one element of the domain of f. If for every element of B there is at least one or more than one element matching with A, then the function is said to be onto.

Complete step-by-step answer:

Before proceeding with the question we should understand the concept of one-one function and onto function.

One to one function basically denotes the mapping of two sets. A function f is one-to-one if every element of the range of f corresponds to exactly one element of the domain of f. One-to-one is also written as 1-1.

If for every element of B there is at least one or more than one element matching with A, then the function is said to be onto
.
The given function $$f(x)=x^3+5x+1$$ maps R to R.

Now let $$y\in R$$ then we get,

$$\begin{array}{rl}& f(x)=y=x^3+5x+1\\ & \Rightarrow x^3+5x+1-y=0.....(1)\end{array}$$

So we can see that the equation (1) is a cubic equation. And we know that any polynomial of odd degree always has at least one real root corresponding to any y belonging to the co-domain, then some x belongs to a domain such that $$f(x)=y$$. Hence, f is onto.

Also we can see that f is continuous on R because it is a polynomial function.

Now differentiating $$f(x)=x^3+5x+1$$ we get,

$$\Rightarrow 3x^2+5......(2)$$

From equation (2) we can see that the polynomial is strictly increasing as $$x^2$$ is always positive. Hence f is one-one also.

Hence $$f(x)=x^3+5x+1$$ is both onto and one-one. Thus the correct answer is option (b).

Note: We have to be very clear with the definitions of one-one function and onto function. Also we have to remember the properties of cubic polynomials and square polynomials which is the key here.