f(x)=4/5x-7 find inverse

Question

how do I find inverse when there is a fraction

 

Answer 1

One does this the same way one does this with finding an inverse for ANY function. Switch x with f(x) (or y) and solve for the new variable...I am not sure if your -7 is in your denominator or if your fraction is supposed to be in front of the -7 and -7 not in the fraction....it is important to delineate which you mean by your parentheses...so I'll answer this question both ways since people can't read your mind...(I say this because it is a common student mistake and may also give conflicting answers when trying to do it on the calculator.)

 

Method 1: (with -7 in denominator)

 

y = 4 / (5x - 7)

x = 4 / (5y - 7)  (Switch x and y)

 

then solve for x

 

x (5y-7) = 4

 

5y - 7 = 4/x

 

5y = 4/x + 7

 

ANSWER: y = (4/x + 7) / 5

 

Method 2: (with the fraction out in front of -7 and -7 not in the fraction)

 

y = (4/5x) - 7  

x = (4/5y) - 7 (switch x and y)

 

Now solve for y

 

x+7 = 4/5y

5y (x+7) = 4

5y = 4 / (x+7)

 

ANSWER: y = 4 /(5(x+7))

Answer 2

f(x)=y=4/5x-7

 

We want to find g(x) = f^-1(x).  The best way to achieve this is to solve for x = f(y)

 

y=4/5x-7 ---> x= g(y)=5/4y+ 35/4

 

To see why this is the inverse, plug x=g(y) into f(x)

 

y= 4/5(5/4y + 35/4) -7 = y  which is what we expected because f(g(x)) gives us back y