Question

If the function f(2x + 3) = 3x- 8, find f -¹(x)​

Answer 1

Explanation:

`let~g(x)=2x+3,~ then~f\circ g(x)=f(g(x)=3x-8\\\\`

lets assume that f(x) is a line, so f(x)=mx+b

`f(g(x))=m(2x+3)+b=2mx+3m+b=3x-8`

the follow equation system results:

`2m=3 \\ 3m+b=-8`

`m=(3)/(2)\\\\b=-(25)/(2)`

finally:

`y=f(x)=(3)/(2)x-(25)/(2)`

now is time to calculate the inverse function:

`y=(3)/(2)x-(25)/(2)=>2y+25=3x=>x=(2)/(3)y+(25)/(3)`

so

`f^(-1)(x)=(2)/(3)x+(25)/(3)`

Answer 2

(2x + 35)/3

Explanation:

f(2x + 3) = 3x - 8

= 3(2x)/2 - 8

= 3(2x + 3 - 3)/2 - 8

= 3(2x + 3)/2 - 3(3)/2 - 8

= ( 3(2x + 3) - 25)/2

Therefore, f(x) = (3x - 25)/2

Let f(x) = (3x - 25)/2 = y

Therefore,

x = (2y + 35)/3 = f(y)

Hence,

f-¹(x) = f(y) = (2y + 35)/3

Just by merely replacing y,

f-¹(x) = (2x + 35)/3