Question

What is the answer ? i proved a picture with the question i need help on

Answer 1

`\begin{gathered} \text{Given} \\ k(x)=(2x)/(3-4x) \end{gathered}`
`\begin{gathered} k(x)=(2x)/(3-4x) \\ y=(2x)/(3-4x) \end{gathered}`

Convert y to x, and x to y

`\begin{gathered} y=(2x)/(3-4x) \\ x=(2y)/(3-4y) \end{gathered}`

Solve for y.

`\begin{gathered} x=(2y)/(3-4y) \\ 3-4y=(2y)/(x) \\ (3-4y)/(2y)=((2y)/(x))/(2y) \\ (3)/(2y)-(4y)/(2y)=(1)/(x) \\ (3)/(2y)-2=(1)/(x) \\ (3)/(2y)=(1)/(x)+2 \end{gathered}`

Get the reciprocal of both sides.

`\begin{gathered} \mleft((3)/(2y)=(1)/(x)+2\mright)^(-1) \\ ((3)/(2y)=(1+2x)/(x))^(-1) \\ (2y)/(3)=(x)/(1+2x) \\ \text{Multiply both sides by }(3)/(2)\text{ to cancel out }(2)/(3)\text{ on the left side} \\ (3)/(2)\mleft((2y)/(3)=(x)/(1+2x)\mright)(3)/(2) \\ y=(3x)/(2(1+2x)) \\ \\ \text{Simplify the denominator} \\ y=(3x)/(2+4x) \\ \text{rearrange the terms on the denominator} \\ y=(3x)/(4x+2) \end{gathered}`
`\begin{gathered} \text{Therefore,} \\ k^(-1)(x)=(3x)/(4x+2) \end{gathered}`