Question

Show that if 𝑎𝑑−𝑏𝑐 ≠ 0, then the function 𝑓(𝑥)=𝑎𝑥+𝑏𝑐𝑥+𝑑 is one-to-one and find its inverse.

Answer 1

If ad-bc=0 then:

 a/c = b/d

y = a/c = b/d

So, y is a constant and is not one-to-one.

Finding the inverse function:

Let y = f(x) = (ax+b) / (cx+d)

So y (cx+d) = ax+b

So cxy + dy = ax+b

So cxy - ax = b - dy

So x (cy - a) = b - dy

So x = (b - dy) / (cy - a)

But x = f^-1(y)

So f^-1(y) = (b - dy) / (cy - a)

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

The fact that we have managed to find the inverse function means that f is one-one.