Question

If f(x)=ax+b and f^-1(x)=bx+a, what are the values of a and b?

Answer 1

`a=-1, b=-1`.

Explanation:

If a function `f(x)` has an inverse function `f^-1(x)`,
`color(red)(f(f^-1(x))=x)` is always true.

Then,
`f(f^-1(x))=x`
`⇔ a(bx+a) +b =x`
`⇔ abx+a^2+b=x`

It must be an identity for `x`. Hence,
`ab=1` (1) and
`a^2+b=0` (2)
are satisfied.

From the equation (2), `b=-a^2`. Substitute this into (1),
`-a^3=1`
`a^3=-1`
`a=-1` (assuming that `a` is a real number.)
`b=-(-1)^2=-1`.