Let f : R → R defined as f (x) = x⁴. Choose the correct answer.
A. f is one-one onto  B. f is many-one onto
C. f is one-one but not onto  D. f is neither one-one nor onto

Solution:

A function f: X → Y is called an onto function if the range of f is Y.

In other words, if each y ∈ Y there exists at least one x ∈ X such that  f(X) = Y,

then f is an onto function.

f : R → R defined as f (x) = x⁴

x, y ∈ R such that f (x) = f (y)

⇒ x⁴ = y⁴

⇒ x = ± y

Therefore,

f ( x) = f ( y ) does not imply that x = y.

For example f (1) = f (- 1) = 1

⇒ f is not one-one.

Consider an element 2 in codomain R there does not exist any x in domain R such that

f (x) = 2.

Therefore,

f is not onto.

Function f is neither one-one nor onto.

The correct answer is D

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.2 Question 11

Let f : R → R defined as f (x) = x⁴. Choose the correct answer. A. f is one-one onto B. f is many-one onto C. f is one-one but not onto D. f is neither one-one nor onto

Summary:

For the function f: R → R defined as f (x) = x⁴, we have shown that the function f is neither one-one nor onto. Hence the correct answer is D