Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by
A = R - {3}, B = R - {1}
f: A → B is defined as.
.
∴ f is one-one.
Let y ∈B = R - {1}. Then, y ≠ 1.
The function f is onto if there exists x ∈A such that f(x) = y.
Now,
Thus, for any y ∈ B, there existssuch that
Hence, function f is one-one and onto.
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
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.
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.
Let f : N → N be defined by
State whether the function f is bijective. Justify your answer.
Let f : R → R be defined as f(x) = 3x. 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.
Determine order and degree(if defined) of differential equation yn + (y')2 + 2y =0
Represent graphically a displacement of 40 km, 30° east of north.
In Figure, identify the following vectors.
(i) Coinitial (ii) Equal (iii) Collinear but not equal
Determine order and degree(if defined) of differential equation yn + 2y' + siny = 0