If f(x)= ax+b, f(2)=1, and f-1(3)=4, find a and b

The value of a and b has to be determined for which if f(x)= ax+b, f(2)=1 and `f^-1(3)=4` .

f(x) = ax + b

f(2) = 1

Now the value of the inverse function `f^-1(x)` is known for x = 3.

If `f^-1(x) = y` , `f(y) = x`

This gives f(4) = 3

This gives two equations that can be solved for a and b

2*a + b = 1 and 4*a + b = 3

2a + b = 1 gives b = 1 - 2a

Substitute this in 4a + b = 3

4a + 1 - 2a = 3

2a = 2

a = 1

As b - 1 - 2a, b = -1

The required value of a and b is a = 1 and b = -1