Question

5. Find the inverses of the following functions:

a. f(x) = (2 / 3 x) − 10
b. g(x) = 2(x + 4)3
c. h(x) = 1 / x−2 , xx ≠ 2

Answer 1

a)

Step 1. We’ll first replace f(x) with y

`y=(2x)/(3)-10`

Step 2. Replace all x’s with y and all y’s with x.

`x=(2y)/(3) -10`

Step 3. Solve for y. Then, replace y with f−1(x)

`x+10=(2y)/(3)\\( 3(x+10))/(2)=y\\ (3x+30)/(2)=y=f^(-1)(x)`

Then,
`f^(-1)(x)=(3x+30)/(2)`

b)

Step 1.
`y=2(x+4)3=6(x+4)=6x+24`

Step 2.
`x=6y+24`

Step 3.

`x-24=6y\\(x-24)/(6)=y=g^(-1)(x)`

Then,
`g^(-1)(x)=(x-24)/(6)`

c)

Step 1.
`y=(1)/(x-2)`

Step 2.
`x=(1)/(y-2)`

Step 3.

`y-2=(1)/(x)\\y=(1)/(x)+2\\h^(-1)(x)=(1)/(x)+2` with
`x\\eq 0`