# Prove that

f is invertible if it is one-one as well as onto.

f(x) = 9x2 + 6x – 5

f’(x) = 18x + 6 > 0 for all x > 0

So, f(x) is one-one.

Now, let y R, then for any x,

f(x) = y if y = 9x2 + 6x – 5

y = (3x)2 + 2(3x) (1) + (1)2 – 5 – 12

y = (3x+1)2 – 6

As x[0,) which means x is a positive real number.

So, x cannot be equal to

Now, fory = - 6

does not belongs to[0,)

Hence, f(x) is not onto

f(x) is not invertible

Now procedure to make it invertible.

Since, x ≥ 0, therefore

Redefining, f: [0, ) [ –5, ) makes f onto

Now, checking if f: [0, ) [ –5, ) is one – one as well as onto.

f(x) = 9x2 + 6x – 5

f’(x) = 18x + 6

f’(x) > 0 for all x [0, )

So, f(x) is one-one in [0, ) … (1)

As, f(x) is an increasing function.

As, x [0, )

So, minimum value of f(x) is at x = 0,

f(0) = 9(0) + 6(0) – 5

f (0) = -5

Also, as x , f(x) .

Also, f(x) is continuous for all x, so f(x) attains all values in

[–5, ).

range of f(x) = co-domain of f(x).

So, f is onto. … (2)

From (1) and (2)-

f(x) is bijective.

Hence f is invertible and f-1: [-5, ) [0, )

OR

Reflexive:

R is reflexive, as 1 + a.a = 1 + a2 > 0

Because square of a number is always positive and 1 added to a positive number is a positive number.

(a, a) R aR

Symmetric:

If (a, b) R then, 1 + ab> 0

1 + ba> 0 (b, a) R

Hence, R is symmetric.

Transitive:

Let a=-8, b=-1,

Since, 1 + ab = 1 + (–8) (–1) = 9> 0

(a, b) R

also,

(b, c) R

but,

Hence, R is not transitive.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses
RELATED QUESTIONS :

Fill in theMathematics - Exemplar

State True Mathematics - Exemplar

State True Mathematics - Exemplar

State True Mathematics - Exemplar

Let A = {1, 2, 3}Mathematics - Exemplar

Show that the relMathematics - Board Papers

Let N denote the Mathematics - Board Papers