Class 12 Relations And Functions · Case Study5 Marksmixed_relations_and_functions
Class 12 Maths · Relations And Functions · Case Study #68
Exhibition swing: the square function
Part (1) 1 Marks
Let \(f:\mathbb{R}\to\mathbb{R}\) be defined by \(f(x)=x^2\). It is
Answer: Option (A): Neither surjective nor injective
- Opposite nonzero inputs have the same image and negative real values are not attained.
Part (2) 1 Marks
Let \(f:\mathbb{N}\to\mathbb{N}\) be defined by \(f(x)=x^2\). It is
Answer: Option (C): Injective
- The square function is strictly increasing on positive natural numbers, but non-squares such as 2 have no preimage.
Part (3) 1 Marks
Let \(f:\{1,2,3,\ldots\}\to\{1,4,9,\ldots\}\) be defined by \(f(x)=x^2\). It is
Answer: Option (A): Bijective
- Each positive square has exactly one positive integer square root. Thus the function is bijective.
Part (4) 1 Marks
Let \(f:\mathbb{N}\to\mathbb{R}\) be defined by \(f(x)=x^2\). Its range is
Answer: Option (A): \(\{1,4,9,16,\ldots\}\)
- The outputs for positive natural inputs are exactly the positive perfect squares.
Part (5) 1 Marks
The function \(f:\mathbb{Z}\to\mathbb{Z}\) defined by \(f(x)=x^2\) is
Answer: Option (A): Neither injective nor surjective
- \(f(1)=f(-1)\), so it is not injective. Negative integers and positive nonsquares have no preimage, so it is not onto.