## Sunday, 13 January 2013

### INVERSE FUNCTION

by Mr Johari

Source: http://www.mathsisfun.com/sets/function-inverse.html

Inverse Function is an optional topic. However it will be great to know and understand.

### QUESTIONS

1. What is the Inverse of an Inverse function?

2. Do all Functions have an inverse? Substantiate your statement.

3. How can we illustrate an Inverse Function graphically?

1. 1) f(x)=2x+3
2x+3=y
x=(y-3)/2
f^-1(x) = (x-3)/2
(f^-1)^-1) = (x-3)/2=y
= x = 2y+3
=2x+3
An inverse of an inverse is the function you started out with

2) No, not all functions have an inverse , for example a f(x)=sinx

3) It mirrors the function and the line of the mirror is y=x

2. Great response

3. 1) f(x)=5x-9
If 5x-9=y , (y+9)/5
Thus , f^-1(x)=(x+9)/5 -> z
So if we inverse this , f^-1(z) = 5x-9 .
So the inverse of an inverse is the original function as you are reversing the process you reversed to get the original

2) No they do not . For some graph in order to form an inverse function , some parts will have to be cut . There is another test we can try . If we draw a horizontal line , and the line cuts through more than 1 point , the function will not have an inverse . So basically , the function is a one to one ..?

3) We take the original graph and if the inverse of the graph is the reflection of the orignal , it is a function . As the angles towards a neutral graph is equal .

4. 1) It is the original function Let us say function y = f(x) = 2x
Domain is x, range is y
Inverse of f(x) : x = f^-1(y) = y/2
another function, g(y) with same relation as the inverse of f.
g^-1(y) = 2y;
therefore the inverse of the inverse of a function is the same as the original function

2)No, for a function that is many to one, a horizontal line.
Let us say we reflect it against y=x. It now becomes a vertical line, and the relationship is one is to many, thus it is not a function

3) Reflect all points on the graph against the line of y = x, thus the original x value of the point is the new y value, and the original y value of the point is the new x value

5. 1) f(x)=4x+6=y
x= (2y-6)/2, f^-1(x)= (2x-6)/2
(f^-1)^-1)=(2x-6)/2=y
=4x+6
The inverse of an inverse is the original function.

2)Not all functions, f(x)=sinx is one example of a function that does not have an inverse.

3) First draw the line y=x and then reflect the points on one side of the line to the other, for example, y=f(x), will become y=f^-1(x) on the other side.

6. 1) f(x) = x + 1 = y
x = y - 1
Therefore, f^-1(x) = y - 1
y - 1 = x
y = x + 1
Therefore, (f^-1)^-1)(x) = x + 1
The inverse of a function is the original function, similar to the index multiplication law in indices where (a^m)^n = a^mn

2) Not all functions have an inverse, f(x) = cos(x) is one example.

3) It is the 'reflection' of the original graph.

7. 1) The inverse of an inverse function is the function itself.
f(x)=3x+4
3x+4=y
x=(y-4)/3
f^-1(x)=(x-4)/3
y=(x-4)/3
x=3y+4
y=3x+4
Therefore the inverse of an inverse function is the function itself.

2) No. There is no inverse for f(x)=sin(x)

3) Take the graph and reflect the points on the line y=x.

8. 1) An inverse of an inverse function is simply just the function itself.
For example, the inverse of f(x)=3x+5 is just f(y)^-1=(x-5)/3. And the inverse of the latter is f(x)=3x+5.
Hence the in inverse of an inverse function is simply just the function itself.

2) Nope. Functions such as f(x) = sin(x) or f(x) = cos(x) do not have an inverse function.

3) The inverse function is a reflection of the original function, with the line of reflection being y=x.

9. The inverse of the inverse function is the initial function

No, f(x)=sin(x) is an example of a function without an inverse function

The inverse function is a reflection of the initial function with x=y as the normal line