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
Well , I do not go the regular equation in the 2nd step.
y = ax+b . The inverse of this finction is got by interchanging x and y and making y the subject of the equation:
y = (x-b)/a.
Putting y=4 and x=3, we get:
So 4 = (3-b)a or
4a =3-b or
4a+b = 3 is the second equation, which is the one we got earlier solution. So have to solve the same set of two equations involving a and b. So go by whichever way you feel easier.
Well, those are the correct answers, however in the second step, you put the inverse answers into the regular equation. Don't you have to put the inverse answers in the inverse equation?
f(x) = ax+b.Therefore,
To detrmine the function or to find the value of a and b under the given conditions that f(2) = 1 and f-1(3) = 4.
f(2) =a*2+b =2a+b =1 by data............................(1)
f-1(3) = f inverse (3)= 4 or f(4) = 3 by data or
f(4) = a*4+b = 4a+b = 3 by data..........................(2)
(2)-(1) eliminates b: (4a+b) - (2a+b) = 3-1=2 or
a=1 . Substituting the value of a in (1), we get,
2*a +b=1 or
b=1-2a =1-2*1 =-1 or
a=1 and b=-1.