Is the function f R R defined by fx 2x 3 if x Q 4x 5 oth

Is the function f : R R, defined by

f(x) = ( 2x 3 if x Q

4x + 5 otherwise

injective (one-to-one)? Is f surjective (onto)?

If x is rational then so is 2x-3

If x is irrational then 4x+5 is also irrational.

Let, f(x)=f(y)

Case 1: x,y both rational

2x-3=2y-3

So, x=y

Case 2: x,y both irrational

4x+5=4y+5

Hence, x=y

Hence, f is one to one

Now check if f is surjective

Case 1: y is rational

Let, 2x-3=y

x=(y+3)/2

(y+3)/2 is also rational

f((y+3)/2)=y

Case 2: y is irrational

Let, 4x+5=y

x=(y-5)/4

(y-5)/4 is also irrational

f((y-5)/4)=y

So , f is onto