Question

consider f:R to R defined by f(x)=5x+3 show that f is a bijective function​

Answer 1

Answer:

First you need to show that the function is one-to-one: put into precise mathematical terms, this means

f(x)=f(y)⇒x=y  

Or in English: if  f(x)=f(y) , then  x=y  

Suppose  f(x)=f(y)  which means

3x−5=3y−5  so  3x=3y which means  x=y .

Therefore,  f(x)  is one-to-one.

Now we need to show  f(x)  is onto  R  

This means that  R=Rng(f) . Now the

range of f is the set of values that f maps to from the

domain. Now, we need to show that

R⊆Rng(f) (we know that

Rng(f)⊆R  since f is a function)

So, suppose  y∈R  and let

x=y+53 . Since  y∈R  

that means  x∈R . Now consider

f(x) .

f(x)=3(y+53)−5  

f(x)=y+5−5=y . This means  y∈Rng(f) ;

thus,  R⊆Rng(f)  and

Rng(f)⊆R  which together

mean  R=Rng(f) ; thus, f is onto.

Since f is one-to-one and onto, it is by definition a

bijection. Q.E.D

Step-by-step explanation:

Answer 2

Answer:

So this is your perfecct answer