Question

Find an equation for this inverse.

Answer 1

Don you mean find its inverse function??
If so, then you can determine if a function has an inverse by passing the "horizontal line test."
This function does because it has a slope not equal to zero. Inverses of horizontal lines have vertical lines in them, which are not by definition functions.
So now, simply replace "f(x)" with "y", then switch the x and y, and solve for y:
y = 5x + 1
x = 5y + 1
-1 -1
5y = x - 1
÷5 ÷5
y = (x-1)/5

Answer 2

The easiest way to find the inverse of a function is to swap the x and y, meaning:

We have the function $f(x)=5x+1$. Let's call our outputs "y", so we have $y=5x+1$. To find the inverse, we're going to change the x and the y's position in the equation, so we have $x=5y+1$. Then all we need to do is solve for y:
$x=5y+1$
$x-1=5y$
$\frac{x}{5}-\frac{1}{5}=y$
$f^{-1}(x)=y=\frac{x}{5}-\frac{1}{5}$