Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f (x) = x – 2/ x – 3 ∀ x ∈ A . Then show that f is bijective.

Dear Student

A= R–{3}, B = R–{1}
And,
f : A → B be defined by f (x) = x –2/ x–3∀x∈A Hence, f (x) = (x–3 + 1)/ (x–3) = 1 + 1/ (x–3) Let f(x1) = f (x2)
1 + 1/x1 –3 = 1+ 1/ x2 –3

1/x1 –3 = 1/ x2 –3
so,
x1=x2

Thus,f (x) is an injective function.
Now let y = (x - 2)/ (x -3)
x–2 = xy–3y
x(1 - y) = 2–3y

x = (3y–2)/ (y - 1)
y∈R–{1} = B
so, f (x) is onto or subjective.
Thus, f(x) is a bijective function

Thanks