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

Question

Lincoln Anderson · Accepted Answer

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

An inverse graph is found by reflecting the original graph in the line y=x. The easiest way to find the inverse function is by setting y=f(x), making x the subject and then switching y and x.

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

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

#xy-3y=2x+1#

#xy-2x=1+3y#

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

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

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

Christian Antunez · Answer

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

1. Replace $ f(x) $ with $ y $.
2. Swap $ x $ and $ y $ to rewrite the equation as $ x = \frac{2y + 1}{y - 3} $.
3. Solve this equation for $ y $, which represents the inverse function.

The steps are as follows:

$$ x = \frac{2y + 1}{y - 3} $$

Multiply both sides of the equation by $ y - 3 $ to eliminate the denominator:

$$ x(y - 3) = 2y + 1 $$

Distribute $ x $ on the left side:

$$ xy - 3x = 2y + 1 $$

Move terms involving $ y $ to one side of the equation and terms without $ y $ to the other side:

$$ xy - 2y = 3x + 1 $$

Factor out $ y $ on the left side:

$$ y(x - 2) = 3x + 1 $$

Divide both sides by $ x - 2 $ to isolate $ y $:

$$ y = \frac{3x + 1}{x - 2} $$

So, the inverse of $ f(x) $ is $ f^{-1}(x) = \frac{3x + 1}{x - 2} $.

Elias Ackerman · Answer

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

1. Replace $ f(x) $ with $ y $: 
$$ y = \frac{2x + 1}{x - 3} $$

2. Swap $ x $ and $ y $:
$$ x = \frac{2y + 1}{y - 3} $$

3. Solve for $ y $:
$$ x(y - 3) = 2y + 1 $$
$$ xy - 3x = 2y + 1 $$
$$ xy - 2y = 3x + 1 $$
$$ y(x - 2) = 3x + 1 $$
$$ y = \frac{3x + 1}{x - 2} $$

4. Replace $ y $ with $ f^{-1}(x) $:
$$ f^{-1}(x) = \frac{3x + 1}{x - 2} $$

So, the inverse of the function $ f(x) = \frac{2x + 1}{x - 3} $ is $ f^{-1}(x) = \frac{3x + 1}{x - 2} $.