How do you find the inverse of #f(x)=(2x+7)/(3x-1)#?

Question

Elijah Allen · Accepted Answer

Let #y=f(x)#.

Eliminate all but one of the #x#'s then rearrange to isolate and find a formula for #x# in terms of #y#. This is the inverse function.

#f^(-1)(y) = (y+7)/(3y-2)# Let #y = f(x) = (2x+7)/(3x-1)# #y = (2x+7)/(3x-1)# I like to reduce the number of occurrences of #x# to one. Then it is clear how to isolate #x# on one side of the equation. The #(2x ...) / (3x ...)# is going to make things a little messy - unless we multiply through by #3# first, so let's do that: Multiply both sides by #3# to get: #3y = (3(2x+7))/(3x-1)# #=(6x+21)/(3x-1)# #=(6x-2+23)/(3x-1)# #=(2(3x-1)+23)/(3x-1)# #=2+23/(3x-1)# Subtract #2# from both sides to get: #3y - 2 = 23/(3x-1)# Multiply both sides by #(3x-1)# to get: #(3y-2)(3x-1) = 23# Divide both sides by #(3y-2)# to get: #3x-1 = 23/(3y-2)# Add #1# to both sides to get: #3x = 23/(3y-2)+1# #=23/(3y-2)+(3y-2)/(3y-2)# #=(23+(3y-2))/(3y-2)# #=(3y+21)/(3y-2)# #=(3(y+7))/(3y-2)# Divide both sides by #3# to get: #x = (y+7)/(3y-2)# So #f^(-1)(y) = (y+7)/(3y-2)#

Anna Allen · Answer

#f^-1(x)=(x+7)/(3x-2)#

Replace #f(x)# with #y#.

#y=(2x+7)/(3x-1)#

Switch all occurrences of #x# with #y# and the #y# with #x#.

#x=(2y+7)/(3y-1)#

Solve for #y#. This may seem daunting, but just start by trying to eliminate fractions by cross-multiplying.

#x(3y-1)=2y+7#

Distribute on the left.

#3xy-x=2y+7#

Get all terms with #y# on one side of the equation. Move anything not including a #y# term to the other side of the equation.

#3xy-2y=x+7#

Factor a #y# term on the left.

#y(3x-2)=x+7#

Divide both sides by #(3x-2)# to solve for #y#.

#y=(x+7)/(3x-2)#

Since this is the inverse function, we can write is with #f^-1(x)# instead of #y#. All #f^-1(x)# means is the inverse of the function #f(x)#.

#f^-1(x)=(x+7)/(3x-2)#

Liam Brennan · Answer

undefined To find the inverse of the function $ f(x) = \frac{2x + 7}{3x - 1} $, follow these steps:

1. Replace $ f(x) $ with $ y $.
2. Swap the roles of $ x $ and $ y $: Rewrite the equation as $ x = \frac{2y + 7}{3y - 1} $.
3. Solve this equation for $ y $.
4. Once you have $ y $ expressed in terms of $ x $, replace $ y $ with $ f^{-1}(x) $ to denote the inverse function.

Let's go through the steps:

1. Original function: $ f(x) = \frac{2x + 7}{3x - 1} $
2. Swap roles of $ x $ and $ y $: $ x = \frac{2y + 7}{3y - 1} $
3. Solve for $ y $:
$$
x(3y - 1) = 2y + 7 \
3xy - x = 2y + 7 \
3xy - 2y = x + 7 \
y(3x - 2) = x + 7 \
y = \frac{x + 7}{3x - 2}
$$
4. So, the inverse function is $ f^{-1}(x) = \frac{x + 7}{3x - 2} $.