What is the inverse of the function () 2x 10?

I assume that we want to find the inverse of the function:

f(x) = 2*x + 10

Remember that the inverse of a function f(x), is a function g(x) such that:

f( g(x) ) = g( f(x) ) = x

Because f(x) is a linear function, we can assume that g(x) will also be a linear function:

g(x) = a*x + b

let's find the values of a and b.

We will have that:

f( g(x) ) = 2*g(x) + 10 = 2*(a*x + b)  + 10

And that must be equal to x, then we need to solve:

2*(a*x + b)  + 10 = x

2*a*x + 2*b + 10 = x

this must be true for all values of x, so we can separate it as:

(2*a*x) + (2*b + 10) = x + 0

2*a*x = x    (one equation for the terms with x)

2*b + 10 = 0

Solving these two equations we get:

2*b = -10

b = -10/2 = -5

2*a*x = x

2*a = 1

a = 1/2

Then the inverse function is:

g(x) = (1/2)*x - 5