SOLUTION: Find the inverse of f(x)=(4x-2)/(3x+2). Verify your answer by showing that f(f^-1(x))=x and f^-1(f(x))=x.

Question 884712: Find the inverse of f(x)=(4x-2)/(3x+2). Verify your answer by showing that f(f^-1(x))=x and f^-1(f(x))=x.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
see the attached worksheet.
step 1 is the original equation.
step 2 interchanges the y and the x.
step 3 solves for y in the interchanged equation.
you get:
f(x) = (4x-2) / (3x + 2) which is the original equation of f(x).
g(x) = (-2x-2) / (3x-4) which is the inverse equation.
you wind up with:
g(x) = f^-1(x) and f(x) = g^-1(x).
this is because:
if a is the inverse of b then b is the inverse of a.

step 4 shows that f(g(x)) is equal to x.
step 5 shows that g(f(x)) is equal to x.

the method of solving f(g(x)) and g(f(x) is to multiply numerator and denominator by the common denominator which removes the fractions and allows you to solve the equations easier.

that's why step 4 equation is multiplied by (3x-4) / (3x-4) and why step 5 equation is multiplied by (3x + 2) / (3x + 2).

see below the pictures for additional comments.
$$$

you get f(g(x)) = x and g(f(x)) = x which confirms that the equations are inverses of each other.

the general idea to find the inverse equatgion is to do steps 1, 2 and 3.
you replace x with y and y with x and then you solve for y.
that's what was done in step 3.

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