Which of the following functions are onetoone I II III IV So

Which of the following functions are one-to-one?

I.
II.
III.
IV.

Solution

I) f(x) = 4x² +2

f(a) = f(b)

==> 4a² +2 = 4b² +2

==> 4a² = 4b²

==> a² = b²

==> a = +b , -b     (Hence not a one - one function)

II) g(x) = 2(x +3)²

g(a) = g(b)

==> 2(a +3)² = 2(b +3)²

==> (a +3)² = (b +3)²

==> a+3 = b+3 , -b+3

==> a = b , -b   (hence not a one - one function)

III) h(x) = 3x³ +3

==> h(a) = h(b)

==> 3a³ +3 = 3b³ +3

==> 3a³ = 3b³

==> a³ = b³

==> a= b (hence one - one function)

IV) k(x) = |4x -3|

==> k(a) = k(b)

==> |4a -3| = |4b -3|

==> 4a -3 = 4b -3 , -(4b -3)

==> a = b , (-4b +6)/4 (hence not a one - one function)

Hence only Option III is one - one function