1. Inverse Functions and their Representations

2. Definition
A function is a set of ordered pairs with no two first elements alike.
f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) }
But ... what if we reverse the order of the pairs?
This is also a function ... it is the inverse function
f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }

3. Example
Consider an element of an electrical circuit which increases its resistance as a function of temperature.
T = Temp
R = Resistance
-20 50
150 20
250 40
350
R = f(T)

4. Now we would say that g(R) and f(T) are inverse functions
Example
We could also take the view that we wish to determine T, temperature as a function of R, resistance.
R = Resistance
T = Temp 50
-20
150
250 20
350 40
T = g(R)
Now we would say that g(R) and f(T) are inverse functions

5. For example, let’s take a look at the square function: f(x) = x2
In the same way, the inverse of a given function will “undo” what the original function did.
For example, let’s take a look at the square function: f(x) = x2 x
f(x) y
f--1(x) 9 3 3 9 9 3 3 9 9 3 3 9 9 3 3 x2 9 9 3 3 9 9 9 3 3 3 9 9

6. For example, let’s take a look at the square function: f(x) = x2
In the same way, the inverse of a given function will “undo” what the original function did.
For example, let’s take a look at the square function: f(x) = x2 x y
f--1(x)
f(x) 5 25 5 5 5 25 25 5 5 25 25 5 5 x2 25 5 5 25 5 25 25 5 25 5 5 5

7. For example, let’s take a look at the square function: f(x) = x2
In the same way, the inverse of a given function will “undo” what the original function did.
For example, let’s take a look at the square function: f(x) = x2 x
f(x) y
f--1(x) 11
121 11 11 11
121
121 11 11
121
121 11 11 x2
121
121
121 11 11
121
121 11 11
121
121
121 11
121 11

8. Graphically, the x and y values of a point are switched.
The point (4, 7)
has an inverse point of (7, 4)
AND
The point (-5, 3)
has an inverse point of (3, -5)

9. Graphically, the x and y values of a point are switched.
If the function y = g(x) contains the points x 1 2 3 4 y 8 16
then its inverse, y = g-1(x), contains the points x 1 2 4 8 16 y 3
Where is there a line of reflection?

10. y = f(x)
y = x
The graph of a function and its inverse are mirror images about the line
y = f-1(x)
y = x

11. Find the inverse of a function :
Example 1: y = 6x - 12
Step 1: Switch x and y:
x = 6y - 12
Step 2: Solve for y:

12. Example 2: Given the function : y = 3x2 + 2 find the inverse:
Step 1: Switch x and y:
x = 3y2 + 2
Step 2: Solve for y:

13. Definition of an Inverse Function
A function, f, has an inverse function, g, if and only if f(g(x)) = x and g(f(x)) = x, for every x in domain of g and in the domain of f.