Question

If f(x)=5x-12, find a value for $x$ so that f^{-1}(x)=f(x+1).

Answer 1

Explanation:

let f(x)=y

y=5x-12

flip x and y

x=5y-12

5y=x+12

`y=(x+12)/(5) \\or \\f^(-1)(x)=(x+12)/(5) \\f(x+1)=5(x+1)-12=5x-7\\(x+12)/(5) =5x-7\\x+12=25x-35\\25~x-x=12+35\\24 x=47\\x=(47)/(24)`

Answer 2

`\boxed{\sf \ \ \ x=(47)/(24) \ \ \ }`

Explanation:

Hello,

f(x)=5x-12

we need to find x so that

`f^(-1)(x)=f(x+1)`

so first of all we can write

`x = (fof^(-1))(x)=f(f^(-1)(x))=5f^(-1)(x)-12\\\\<=>5f^(-1)(x)=x+12\\\\<=>f^(-1)(x)=(x+12)/(5)`

and f(x+1) = 5(x+1) - 12 = 5x + 5 -12 = 5x - 7

then solving

`f^(-1)(x)=f(x+1)`

is equivalent to

`(x+12)/(5)=5x-7 \ multiply \ by \ 5 \\\\<=> x+12 = 5(5x-7)=25x-35\\<=> 25x-x=12+35\\\\<=>24x=47\\\\<=>x=(47)/(24)`

Hope this helps