Question

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

Answer 1

Answer:

Step-by-step 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:

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

$m=\frac{3}{2}\\\\b=-\frac{25}{2}$

finally:

$y=f(x)=\frac{3}{2}x-\frac{25}{2}$

now is time to calculate the inverse function:

$y=\frac{3}{2}x-\frac{25}{2}=>2y+25=3x=>x=\frac{2}{3}y+\frac{25}{3}$

so

$f^{-1}(x)=\frac{2}{3}x+\frac{25}{3}$

Answer 2

Answer:

(2x + 35)/3

Step-by-step 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