Question

How do I solve and what would the answer be?

Answer 1

Given the question in the image, the following are the solution steps to get the inverse of the function

STEP 1: Write the given function

`f(x)=(2)/(x-5)`

STEP 2: Define an inverse of a function

`\mathrm{A\: function\: g\: is\: the\: inverse\: of\: function\: f\: if\: for}\: y=f\mleft(x\mright),\: \: x=g\mleft(y\mright)\:`

STEP 3: Find the inverse of the given function

`\begin{gathered} f(x)=(2)/(x-5) \\ \text{Set the function f(x) to y} \\ y=f(x)=(2)/(x-5) \\ y=(2)/(x-5) \\ \text{Swap x with y} \\ x=(2)/(y-5) \\ \text{solve for y} \\ By\text{ cross multiplication,} \\ x(y-5)=2 \\ xy-5x=2 \\ \text{Add 5x to both sides} \\ xy-5x+5x_{}=2+5x \\ xy=2+5x \\ \text{Divide both sides by x} \\ (xy)/(x)=(2+5x)/(x) \\ y=(2+5x)/(x)=(2)/(x)+(5x)/(x) \\ y=(2)/(x)+5 \\ \text{Set the inverse to y} \\ f^(-1)(x)=_{}(2)/(x)+5 \end{gathered}`

Hence, the inverse of the function is;

`(2)/(x)+5`