# Given #f(x) = (3-2x) / (2x+1)# and #f(g(x)) = 7 - 3x# how do you find g(x)?

##### 1 Answer

#### Explanation:

Even though we don't know

#f(x) = (3 - 2x) / (2x + 1) " "=> " " f(g(x)) = (3 - 2 g(x)) / (2 g(x) + 1)#

We also know that

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

Let me write

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

Now, you need to solve this equation for

... multiply both sides with

#<=> 3 - 2g = (7 - 3x) * (2g + 1)#

#<=> 3 - 2g = (7 - 3x) * 2g + (7 - 3x)#

Bring all products that include

So, subtract

#<=> - 2g - (7 - 3x) * 2g = (7 - 3x) - 3#

... factorize

#<=> (-2 - 14 + 6x) * g = 4 - 3x#

#<=> (-16 + 6x) * g = 4 - 3x#

... divide both sides by

#<=> g = (4 - 3x)/(-16 + 6x) = (4 - 3x)/(2(-8 + 3x))#

Thus, we have

#g(x) = (4 - 3x) / (-16 + 6x)#

It might be a good idea to test if the calculation was correct. To do so, compute

#f(g(x)) = f((4 - 3x) / (-16 + 6x)) #

#= (3 - 2 * (4 - 3x) / (2(-8+ 3x)))/(2 * (4 - 3x) / (2(-8 + 3x)) + 1) #

#= (3 - (4 - 3x) / (-8 + 3x))/( (4 - 3x) /(-8+ 3x) + 1) #

#= ((3(-8 + 3x) - (4 - 3x))/(-8 + 3x)) / ((4 - 3x + (-8 + 3x))/(-8 + 3x))#

#= (3(-8 + 3x) - (4 - 3x)) / (4 - 3x + (-8 + 3x)) = (-28 +12x) / (-4)#

#= 7 - 3x#