NCERT Class 12 · Ch 1 Exercise 1.2 · Q1

NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.2 Question 1

Show that f:ℝ*→ℝ*, f(x)=1/x, is one-one and onto, where ℝ*=ℝ∖{0}. Does the conclusion remain true if the domain is ℕ and codomain remains ℝ*?

💡 Concept Hints

For onto, start with an arbitrary element of the declared codomain and solve f(x)=y.
Separate the one-one and onto claims. For onto, ensure each inverse candidate lies in the allowed domain.

Final Answer: Bijective on ℝ*; with domain ℕ it is injective but not onto ℝ*.

If f(a)=f(b), then 1/a=1/b. Since a,b≠0, multiplication by ab gives a=b, so f is injective.
For every y∈ℝ*, choose x=1/y∈ℝ*. Then f(x)=y, so f is onto.
On ℕ the same equality argument proves injectivity, but every output is positive. For example −1∈ℝ* has no preimage in ℕ.
Thus the restricted-domain function is not bijective.

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