Question

The function F(x)=6/x is one to oneFind parts A and B

Answer 1

(a)

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

Step-by-step explanation

Given function:

`f(x)=(6)/(x)`

(a) To find f⁻¹(x)

`\begin{gathered} \text{Let y }=f(x) \\ \text{This implies, y }=(6)/(x) \\ x=(6)/(y) \\ \text{Note that x }=f^(-1)(y) \\ f^(-1)(y)=(6)/(y) \\ \therefore f^(-1)(x)=(6)/(x) \end{gathered}`

(b) To show that

`f(f^(-1)(x))=x\text{ and }f^(-1)(f(x))=x`
`\begin{gathered} To\text{ find }f(f^(-1)(x)) \\ \text{Subtitute }x=f^(-1)(x)\text{ into }f(f^(-1)(x)) \\ \text{Since }f(x)=(6)/(x),\text{ it follows that} \\ f(f^(-1)(x))=(6)/(f^(-1)(x))=6/(6)/(x)=6*(x)/(6)=x \end{gathered}`

Also,

`\begin{gathered} to\text{ find }f^(-1)(f(x)) \\ \text{Substitute }x=^{}f(x)\text{ into }f^(-1)(f(x)) \\ \text{Since }f^(-1)(x)=(6)/(x),\text{ it follows that} \\ f^(-1)(f(x))=(6)/(f(x))=6/(6)/(x)=6*(x)/(6)=x \end{gathered}`

Therefore, it is verified that

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