Please help with this problem

Question

Let f(x) =4x+2/3x-6

a) find f^-1, the inverse function of f

B) what is the domain of f? What is the domain of the inverse function?

C) what is f(3)?

D) what is f^-1, where the number in the blink is your answer from part (c)

Answer 1

f^-1(x) is the inverse function. let y=f^-1(x) = the inverse of (4x+2)/(3x-6). switch x for y

everywhere there's an x, replace it by y, where there's a y replace it by x

to get:

x=(4y-2)/(3y-6) Solve for y and you get the inverse function

Cross multiply

x(3y-6)= 4y-2

3xy-6x = 4y-2

put the y terms together

3xy-4y = 6x -2

factor out the y

y(3x-4) = 6x - 2

divide by 3x-4

y = (6x-2)/(3x-4) that's the inverse function

f^-1(x) = (6x-2)/(3x-4)

domain of f(x)? all values of x that don't make the denominator zero. 3x-6 can't equal 0

3x-6=0 means 3x=6 or x=2. x cannot equal 2. The domain of x is all real numbers except x=2

In interval notation, that's (-infinity, 2) and (2,infinity)

f(3)? just plug in 3 for wherever x appears

f(3)= (4(3)+2))/(3(3)-6)= 14/3

f^-1(3)? Take the f^-1(x) function and plug in 3

to get:

f^-1(3)=[6(3)-2]/[3(3)-4] = (18-2)/(9-4) = 16/5

f^-1(3) = 16/5