For this case, we must find the inverse of the following function:
[tex]f (x) = \frac {5x + 1} {- x + 7}\\x\neq 7[/tex]
To find the inverse we follow the steps below:
[tex]y = \frac {5x + 1} {- x + 7}[/tex]
We rewrite the denominator:
[tex]y = \frac {5x + 1} {- (x-7)}\\y = - \frac {5x + 1} {(x-7)}[/tex]
We exchange variables:
[tex]x = - \frac{5y + 1} {(y-7)}[/tex]
We solve for y:
We multiply on both sides of the equation by (y-7)
[tex]x (y-7) = - 5y-1\\xy-7x = -5y-1[/tex]
We subtract xy on both sides of the equation:
[tex]-7x = -5y-1-xy[/tex]
We add 1 to both sides:
[tex]-7x + 1 = -5y-xy[/tex]
We factor for y:
[tex]-7x + 1 = y (-5-x)[/tex]
We divide both sides by (-5-x):
[tex]y = \frac {-7x + 1} {- 5-x}[/tex]
So, we have:
[tex]f ^ {- 1} (x) = \frac {-7x + 1} {- 5-x}[/tex]
Answer:
[tex]f ^ {- 1} (x) = \frac {-7x + 1} {- 5-x}[/tex]
[tex]x\neq -5[/tex]
