Question

A function f:R→R defined by f(x)=5x^4+2 is one – one but not onto.

(a) True

(b) False

I have been asked this question in an international level competition.

This key question is from Types of Functions topic in division Relations and Functions of Mathematics – Class 12

Answer 1

Correct choice is (b) False

The explanation is: The above statement is false. f is neither one-one nor onto.

For one-one: Consider f(x1)=f(x2)

∴ 5x1^4+2=5x2^4+2

⇒x1=± x2.

Hence, the function is not one – one.

For onto: Consider the real number 1 which lies in co- domain R, and let \(x=(\frac{y-2}{5})^{\frac{1}{4}}\).

Clearly, there is no real value of x which lies in the domain R such that f(x)=y.

Therefore, f is not onto as every element lying in the codomain must have a pre- image in the domain.