Problem 11 Solution

This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.2 Problem 11 Solution. Solutions for other problems are available at Exercise 1.2 Solutions
Exercise 1.2 Problem 11 Solution
11. Let f : R → R be defined as {f(x) = x^4}. Choose the correct answer.
f is one-one onto
f is many-one onto
f is one-one but not onto
f is neither one-one nor onto.
To Check whether f is one-one:
Consider two elements x_1, x_2R such that
{f(x_1) = f(x_2)}
{x_1^4 = x_2^4}
{x_1^2 = x_2^2}
{x_1 = \pm x_2}
⇒ Two different elements in the domain have the same image in the co-domain.
f is not one-one.
To Check whether f is onto:
For f to be onto, every element yR in the co-domain should be an image of some element xR in the domain, such that
{y = f(x) = x^4}
As we know, x^4 is always positive.
However, as the function f is defined as f : R → R, the co-domain, which is set of real numbers has both positive and negative numbers, will have only the positive numbers as the images of elements in the domain. In otherwords, the negative numbers will not be images of any elements in the domain.
f is not onto.
∴ As f is neither one-one nor onto, D is the correct answer.