Prove that f(x)=-2x+3 is inveritble. If f is invertible, find f^-1.Find the point of the graph of f and f^-1.

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:

-2x1 + 3 = -2x2 +
3

We'll eliminate like
terms:

-2x1 = -2x2

We'll
divide by -2:

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 =
-2x + 3

We'll isolate x to one side. For this reason, we'll
add -3 both side:

y - 3 =
-2x

We'll use the symmetric
property:

-2x = y-3

We'll
divide by -2:

x = (3-y)/2

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) =
(3-x)/2