• NCERT Chapter
Question 1

# Show that the function f : R* → R* defined by f(x) = 1/x is one-one and onto,where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R* ?

It is given that f: R* → R* is defined by

One-one:

f is one-one.

Onto: It is clear that for y∈R*, there exists such that

∴f is onto.

Thus, the given function (f) is one-one and onto.

Now, consider function g: N → Rdefined by

We have,

g is one-one.

Further, it is clear that g is not onto as for 1.2 ∈R* there does not exit any x in N such that g(x) =.

Hence, function g is one-one but not onto.

">

Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by

• Q:-

Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

• Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b2} is neither reflexive nor symmetric nor transitive.
• Q:-

Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

• Q:- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
• ## Recently Viewed Questions of Class 12th mathematics

">

Let A = R – {3} and B = R – {1}. Consider the function  f : A → B defined by

• Q:-

A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

• Q:-

Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.