Question

Find an Equation for inverse function

f(x)=x+9 one to one

f^-1(x)=

Answer 1

Inverse Functions
An inverse function goes in the opposite direction!

Let us start with an example:

Here we have the function f(x) = 2x+3, written as a flow diagram:



The Inverse Function just goes the other way:



So the inverse of: 2x+3 is: (y-3)/2
The inverse is usually shown by putting a little "-1" after the function name, like this:

f-1(y)

We say "f inverse of y"

So, the inverse of f(x) = 2x+3 is written:

f-1(y) = (y-3)/2

(I also used y instead of x to show that we are using a different value.)

Back to Where We Started
The cool thing about the inverse is that it should give you back the original value:


If the function f turns the apple into a banana,
Then the inverse function f-1 turns the banana back to the apple


Example:
Using the formulas from above, we can start with x=4:
f(4) = 2×4+3 = 11
We can then use the inverse on the 11:
f-1(11) = (11-3)/2 = 4
And we magically get 4 back again!
We can write that in one line:
f-1( f(4) ) = 4
So applying a function f and then its inverse f-1 gives us the original value back again:

f-1( f(x) ) = x

We could also have put the functions in the other order and it still works:

f( f-1(x) ) = x

Example:
Start with:
f-1(11) = (11-3)/2 = 4
And then:
f(4) = 2×4+3 = 11
So we can say:
f( f-1(11) ) = 11


Solve Using Algebra
You can also work out the inverse using Algebra. Put "y" for "f(x)" and solve for x:

The function: f(x) = 2x+3
Put "y" for "f(x)": y = 2x+3
Subtract 3 from both sides: y-3 = 2x
Divide both sides by 2: (y-3)/2 = x
Swap sides: x = (y-3)/2

Solution (put "f-1(y)" for "x") : f-1(y) = (y-3)/2

Answer 2

Four steps.

1) Let f(x) = y. In this case, f(x)=x+9=y

2) f^-1(x) is implied by x=y+9 (switch x and y).

3) Solve for y. y=x-9

4) Rename y to f^-1(x) "f inverse of x" y=x-9=f^-1(x)