Please prove with all details,process! Define a function f : R rightarrow R by I

Question

Please prove with all details,process!

Define a function f : R rightarrow R by Is f one-to-one? Is f onto? Is f-1 a function? State and prove a theorem. Let a, b, c and be real numbers. Suppose cd 0 and ad - bc 0. Define f : R {-d/c} rightarrow R {a/c} by f(x) = ax + b/cx + d. Show that f is one-to-one and onto. Give a formula for f-1(x). Let A, B and C be sets. Let f rightarrow B and g : B rightarrow C. Prove that if f and g are onto, then g is onto. Prove that if g o f is onto, then g is onto. If g o f is onto, is f necessarily onto? Prove your answer.

Explanation / Answer

1.

let f(x1) = f(x2)

if x1, x2 are both rational => 2x1= 2x2 => x1 = x2

if x1, x2 are both irrational => -3x1 = -3x2 =. x1 = x2

if x1 is rational, x2 is irrational => 2x1 = -3x2 => x2 = -2x1/3 => x2 is rational => contradiction

if x1 is irrational, x2 is rational => -3x1 = 2x2 => x1 = -2x2/3 => x1 is rational => contradiction

thus f(x1)= f(x2) => x1 = x2 => f is one-one

let x belongs to Q then f(x/2) = x

let y belongs to R-Q, then f(-x/3) =x

=>

f is onto

thus proved

2.

let f(x1) = f(x2)

=>

(ax1+b)/(cx1+d) = (ax2+b)(cx2+d)

=>

(ax1+b)(cx2+d) = (ax2+b)(cx1+d)

=>

(bc-ad)(x1-x2) = 0

=>

x1 = x2

=>

f is one-one

f[(dx-b)/(a-cx)] = x

=>

f is onto

(b)

f^(-1)(x) = (dx-b)/(a-cx)

3.

(a)

let f,g are onto

let x belongs to C, then there exists a x1 belongs to B such that g(x1) = x

f is onto and x1 belongs to B => there exists a x2 belongs to A such that f(x2) = x1

=>

g(f(x2)) = x => gof is onto

(b)

let gof is onto

=>

let x beongs to C =>

there exists a x2 belongs to gof(x2) = x => g[f(x2)] = x

=>

f(x2) belongs to B

=>

for every x belonging to C there exists a x1 =f(x2) belonging to B such that g(x1) =x

=>

g is onto

(c)

consider the following example:

A = {1}, B = (1,2) , C = {3}

f(1) = 2, g(2) = 3 => fog(1) = 3

=> fog is onto but f is not onto.

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