What is the inverse of f(x)=3x+1

The inverse of the function f(x) is
f^-1(x).

To prove that f(x) is invertible, we'll have to
prove first that f(x) is bijective.

To prove that f(x) is
bijective, we'll have to prove that is one-to-one and on-to
function.

1) One-to-one
function.

We'll suppose that f(x1) =
f(x2)

We'll substitute f(x1) and f(x2) by their
expressions:

3x1 + 1= 3x2 +
1

We'll eliminate like
terms:

3x1 = 3x2

We'll divide
by 3:

x1 = x2

A function is
one-to-one if and only if for x1 = x2 => f(x1) =
f(x2).

2) On-to function:

For
a real y, we'll have to prove that it exists a real x.

y =
3x + 1

We'll isolate x to one side. For this reason, we'll
add -1 both side:

y - 1 =
3x

We'll use the symmetric
property:

3x = y - 1

We'll
divide by 3:

x = (y - 1)/3

x
is a real number.

From 1) and 2) we conclude that f(x) is
bijective.

If f(x) is bijective => f(x) is
invertible.

f^-1(x) = (x -
1)/3