Question

Find the inverse of the given function

Answer 1

For this case, we must find the inverse of the following function:

$f (x) = \frac {5x + 1} {- x + 7}\\x\neq 7$

To find the inverse we follow the steps below:

$y = \frac {5x + 1} {- x + 7}$

We rewrite the denominator:

$y = \frac {5x + 1} {- (x-7)}\\y = - \frac {5x + 1} {(x-7)}$

We exchange variables:

$x = - \frac{5y + 1} {(y-7)}$

We solve for y:

We multiply on both sides of the equation by (y-7)

$x (y-7) = - 5y-1\\xy-7x = -5y-1$

We subtract xy on both sides of the equation:

$-7x = -5y-1-xy$

We add 1 to both sides:

$-7x + 1 = -5y-xy$

We factor for y:

$-7x + 1 = y (-5-x)$

We divide both sides by (-5-x):

$y = \frac {-7x + 1} {- 5-x}$

So, we have:

$f ^ {- 1} (x) = \frac {-7x + 1} {- 5-x}$

Answer:

$f ^ {- 1} (x) = \frac {-7x + 1} {- 5-x}$

$x\neq -5$