Answer to Question #191558 in Real Analysis for Amara

Question

 by mathemarical induction on n that

3^n ≥ 2n^2 + 1 for all n ∈ N

Given the function g : R → R defined by

g (x) = {x-1/2x+4 if x̸=−2 and 1/2 x=−2

Find whether or not f is injective and surjective. Find the inverse of f, if it exists.

Answer 1

Here two questions include in single question

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Question1:- by mathematical induction on n that

3^n ≥ 2n^2 + 1 for all n ∈ N

Solution 1:-

3ⁿ ≥ 2n² + 1

First we check it for n=1

Then we get

3≥2+1

3=3

Which is true.

Now we check it for n=2

Then we get

9≥8+1

9=9

Which is also true.

Now we assume it is true for n=k.

Such that

3^k≥2k²+1..........(1)

Now we show that it is also true for k+1.

So put n=k+1 in question.

3^k+1 ≥ 2(k+1)² +1

3^k.3¹ ≥ 2(k²+2k+1)+1

Put value from eq(1)

(2k²+1).3 ≥ 2(k²+2k+1)+1

Simplifying

4k²≥ 4k

k≥ 1

It is true.

hence we say that

3ⁿ ≥ 2n² + 1 .........Hence proved

-------------------------------------------------------------question2:-

g (x) = {x-1}/{2x+4} is injective or surjective?

solution:-

(A)Injective:-

concept for check injective:-If any two x and y are equal, g(x) and g(y) are also equal, then g(x) is injective.

Start to check injective with this concept,

Let

x=y

x-1=y-1.....(1) (substract 1 both side)

Again x=y

2x+4=2y+4...(2) (multiply 2 and add4 both side)

Divide equation 1 by 2

(x-1)/(2x+4) =(y-1)/(2y+4)

g(x)=g(y)

Hence g(x) is injective.

(B)Surjective:-

concept for check surjective:-A function is surjective or onto if each element of the codomain is mapped to by at least one element of the domain.

now Checking to it, so find value of x from question.

g (x) = {x-1}/{2x+4}

g(x) .(2x+4)=(x-1).... (cross multiply of (2x+4))

(g.2x+g.4)=(x-1)...here we write g(x) =g

x(2g-1)=(-1-4g) .....transformation x and 4g

x=(1+4g)/(1-2g)

if we put g=(1/2) then we get x=infinity

,So we say each element of codomain not map on domain.so function is not surjective.

(C)Inverse: if function is injective and surjective both.then function has inverse. So in above condition, function has no inverse.

Note for (C): if g/=(1/2) then function is surjective and then its inverse exist.

And inverse is

(1+4x)/(1-2x)