# 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.

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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**

f(x) = -2x+3.

We see that f(x) = -2x+3 maps R-->R.

f(x) is an onto function as any real number is the image of the function.

Also ff(x) = -2x+3 maps f: R--> R as all diffrent elements the domain R has different images.

If possible x1 unequal to x2 , let let f(x1)= f(x2).

Then -2x1+3 = -2x2 + 3.

Thar implies -2x1 = -2x2. Or

Implies 2(x1-x2) = 0.

Implies = x1-x2 = 0.

Thus x1 unequal to x2 is wrong.

Thet proves f(x) = -2x+3 is both onto and One-one function or a bijection

Therefore if f(x) = y = -2x+3, then -2x= y-3.

x =( y-3)/-2

x = (3-y)/2 s the inverse of y = -2x+3.