Question

Which equation is the inverse of (x-4)^2-2/3=6y-12

Answer 1

Answer-

$The \ inverse \ of \ (x-4)^{2}-\frac{2}{3} =6y-12 \ is \ y= 4\pm \sqrt{6x-\frac{34}{3}}$

Solution-

The given function,

$(x-4)^{2}-\frac{2}{3} =6y-12$

The inverse of a function normally means switching the role of the variables. ( y becomes the input or independent variable, and x becomes the output or the dependent variable)

Switching x and y, the function becomes,

$(y-4)^{2}-\frac{2}{3} =6x-12$

$\Rightarrow (y-4)^{2}= 6x-12 +\frac{2}{3}=6x-\frac{34}{3}$

$\Rightarrow (y-4)= \pm \sqrt{6x-\frac{34}{3}}$

$\Rightarrow y= 4\pm \sqrt{6x-\frac{34}{3}}$

Answer 2

Answer:  $y = \pm \sqrt{\frac{18x-34}{3}} + 4$

Step-by-step explanation:

Here the given function is,

$(x-4)^2-\frac{2}{3} = 6 y - 12$

For finding the inverse of the given function, First we interchange x and y and after that we will isolate y,

By interchanging x and y,

We get,

$(y-4)^2-\frac{2}{3} = 6 x - 12$

$\implies (y-4)^2= 6 x - 12+\frac{2}{3}$

$\implies (y-4)^2=\frac{18x-36+2}{3}$

$\implies (y-4)^2=\frac{18x-34}{3}$

$\implies y-4=\pm \sqrt{\frac{18x-34}{3}}$

$\implies y=\pm \sqrt{\frac{18x-34}{3}}+4$

Which is the required equation of the inverse of the given function.