Question

Which pair of funtions is not a pair of inverse functions? please help!!

Answer 1

`f(x)=(x)/(x+20) , g(x)=(20x)/(x-1)`

Explanation:

we know that

To find the inverse of a function, exchange variables x for y and y for x. Then clear the y-variable to get the inverse function.

we will proceed to verify each case to determine the solution of the problem

case A)
`f(x)=(x+1)/(6) , g(x)=6x-1`

Find the inverse of f(x)

Let

y=f(x)

Exchange variables x for y and y for x

`x=(y+1)/(6)`

Isolate the variable y

`6x=y+1`

`y=6x-1`

Let

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

`f^(-1)(x)=6x-1`

therefore

f(x) and g(x) are inverse functions

case B)
`f(x)=(x-4)/(19) , g(x)=19x+4`

Find the inverse of f(x)

Let

y=f(x)

Exchange variables x for y and y for x

`x=(y-4)/(19)`

Isolate the variable y

`19x=y-4`

`y=19x+4`

Let

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

`f^(-1)(x)=19x+4`

therefore

f(x) and g(x) are inverse functions

case C)
`f(x)=x^(5), g(x)=\sqrt[5]{x}`

Find the inverse of f(x)

Let

y=f(x)

Exchange variables x for y and y for x

`x=y^(5)`

Isolate the variable y

fifth root both members

`y=\sqrt[5]{x}`

Let

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

`f^(-1)(x)=\sqrt[5]{x}`

therefore

f(x) and g(x) are inverse functions

case D)
`f(x)=(x)/(x+20) , g(x)=(20x)/(x-1)`

Find the inverse of f(x)

Let

y=f(x)

Exchange variables x for y and y for x

`x=(y)/(y+20)`

Isolate the variable y

`x(y+20)=y`

`xy+20x=y`

`y-xy=20x`

`y(1-x)=20x`

`y=20x/(1-x)`

Let

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

`f^(-1)(x)=20x/(1-x)`

`(20x)/(1-x)\\eq (20x)/(x-1)`

therefore

f(x) and g(x) is not a pair of inverse functions