1. Warm-up:
Given f(x) = 2x and g(x) = x2 – 3
Find (f ° g)(x)

2. FUNCTIONS and INVERSE FUNCTIONS

3. Functions
Imagine functions are like the dye you use to color eggs. The white egg (x) is put in the function blue dye B(x) and the result is a blue egg (y).

4. The Inverse Function “undoes” what the function does.
The Inverse Function of the BLUE dye is bleach.
The Bleach will “undye” the blue egg and make it white.

5. For example, let’s take a look at the cube function: f(x) = x3
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 cube function: f(x) = x3
f(x) y x
f -1(x)
125 5 5 5 5
125
125 5 5 5 5
125 x3
125 5 5 5
125
125 5 5 5

6. Remember we talked about functions---taking a set X and mapping into a Set Y1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 10 5
Set X
Set Y
An inverse function would reverse that process and map from SetY back into Set X

7. How do you determine that a function has an inverse?
It must either be a
one-to-one function OR
a restricted many-to-one function.

8. How do you determine that a function is one-to one?
Look at values of function
Use a map
Use a set of ordered pairs
Use a table of values
Use the function equation
Horizontal Line Test

9. One-to-One Functions
Only functions that are one-to-one have inverses.
A function is one-to-one if each value in its domain is assigned a different value in the range.

10. If we map what we get out of the function back, we won’t always have a function going back.1 2 2 4 3 6 4 8 5
Since going back, 6 goes back to both 3 and 5, the mapping going back is NOT a function These functions are called many-to-one functions
Only functions that pair the y value (value in the range) with only one x will be functions going back the other way. These functions are called one-to-one functions.

11. This would not be a one-to-one function because to be one-to-one, each y would only be used once with an x. 1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 5 10
This is a function IS one-to-one. Each x is paired with only one y and each y is paired with only one x
Only one-to-one functions will have inverse functions, meaning the mapping back to the original values is also a function.

12. How do you determine that a function is one-to one?3 2 5 9 1 7
Use a mapping diagram. f
ordered pairs (2,1) (3, 7) (5, 3) (9, 0)
map y to x (1, 2) (7, 3) (3, 5) (0, 9) -2 2 5 9 25 4 81 g
One-to-one
ordered pairs (2,4) (-2, 4) (5, 3) (9, 0)
Map y to x (4, 2) (4, -2) (3, 5) (0, 9)
Note g is not one-to-one.

13. How do you determine that a function is one-to one?
Use a table of values.
Given the function at the right
Can it have an inverse?
Why or Why Not?
NO … when we reverse the ordered pairs, the result is Not a function
We would say the function is not one-to-one
A function is one-to-one when different inputs always result in different outputs x Y 1 5 2 9 4 6 7

14. How do you determine that a function is one-to one?
Use a list of ordered pairs.
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) }
g(x) = { (x,y) : (2, 5), (6, 1), (-2, 5), (12, 9) }
Is the function one-to-one?

15. How do you determine that a function is one-to one?
Recall that to determine by the graph if an equation is a function, we have the vertical line test.
If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function.
This is NOT a function
This is a function
This is a function

16. How do you determine that a function is one-to one?
To be a one-to-one function, each y value could only be paired with one x. Let’s look at a couple of graphs.
Look at a y value (for example y = 3)and see if there is only one x value on the graph for it.
For any y value, a horizontal line will only intersection the graph once so will only have one x value
This is a many-to-one function
This then IS a one-to-one function

17. How do you determine that a function is one-to one
How do you determine that a function is one-to one? Use the Horizontal Line Test
Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions. (a) (b)
If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one and therefore has an inverse.
Not one-to-one
One-to-one

18. How do you determine that a function is one-to one?
Horizontal line test?

19. How do you determine that a function is one-to one?
Summary for
Horizontal Line Test
Used to determine whether a function is one-to-one or many-to-one.
Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test.
If the original function passes the horizontal line test, then its inverse is a function.
If the original function does not pass the horizontal line test, then its inverse is not a function.

20. How do you determine that a function is one-to one?
Example Decide whether the function is one-to-one. (a) (b) Solution (a) For this function, two different x-values produce two different y-values. (b) If we choose a = 3 and b = –3, then 3  –3, but

21. How do you find the Inverse Function?
Notation for the inverse function f -1 is read
“f-inverse”
{(-3, 7), (-2, 10), (-1, 12), (0, 15), (1, 10)}
Inverse:
{(-5, 12), (-2, -8), (4, 10), (6, 14), (10, 18)}
** Remember functions have to pass the vertical line test!
***Remember for functions to have an inverse function they have to pass the horizontal line test!

22. How do you find an Equation for the Inverse Function?
Notation for the inverse function f -1 is read
“f-inverse”
Finding the Equation of the Inverse of y = f(x)
1. Interchange x and y.
2. Solve for y.
3. Replace y with f -1(x).
Any restrictions on x and y should be considered.
** Remember functions have to pass the vertical line test!
***Remember for functions to have an inverse function they have to pass the horizontal line test!

23. Given f (x) = -2x – 7, find the inverse
How do you find an equation for the inverse function?
Finding the Equation of the Inverse of y = f(x)
1. Interchange x and y.
2. Solve for y.
3. Replace y with f -1(x).
Any restrictions on x and y should be considered.
Given f (x) = -2x – 7, find the inverse
y = -2x – 7
1. x = -2y – 7
2. y = x + 7 -2
f -1(x) = x + 7

24. Ex: (a)Find the inverse of f(x)=x5.
How do you find an equation for the inverse function?
Ex: (a)Find the inverse of f(x)=x5.
(b) Is f -1(x) a function?
(hint: look at the graph!
Does it pass the vertical line test?)
y = x5
x = y5
Yes , f -1(x) is a function.

25. How do you find an equation for the inverse function?
Ex: f(x)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation.
y = 2x2-4
x = 2y2-4
x+4 = 2y2 OR
f -1(x) is not a function.
*Later we will talk about how you can make this a function.

26. Ex: g(x)=2x3 How do you find an equation for the inverse function?
y=2x3
x=2y3
Inverse is a function!

27. Find the inverse, if it exists, of
How do you find an equation for the inverse function?
Find the inverse, if it exists, of
f (x) = 3x2 + 2
Step 1: Switch x and y:
x = 3y2 + 2
Step 2: Solve for y:

28. How do you find an equation for the inverse function?
Find the inverse, if it exists, of Solution
Write f (x) = y.
Interchange x and y.
Solve for y.
Replace y with f -1(x).

29. How do you find an equation for the inverse function?
Find the inverse, if it exists, of
f(x) = -3x+6

30. How do you find an equation for the inverse function?
Find the inverse, if it exists, of

31. How do you find an equation for the inverse function?
Find the inverse, if it exists, of

32. Let’s look back at your warm-up functions. Do they have an inverse
Let’s look back at your warm-up functions. Do they have an inverse? If so, find the inverse equation.
f(x) = 2x g(x) = x2 – 3

33. What do we know about the graphs of functions and their inverses?
Graph, the x and y values of the following function and it’s inverse.
Suggestion: Use two different colors.
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?
Warm-up

34. What do we know about the graphs of functions and their inverses?
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)
Can also use homework problems p. 265/#27 &29

35. Let’s consider the function and compute some values and graph them.
Notice that the x and y values traded places for the function and its inverse.
These functions are reflections of each other about the line y = x
Let’s consider the function and compute some values and graph them.
x f (x)
(2,5)
(5,2)
x f -1(x)
(-3,-2)
Let’s take the values we got out of the function and put them into the inverse function and plot them
(-2,-3)
Yes, so it will have an inverse function
Is this a one-to-one function?

36. Let’s consider the function and compute some values and graph them.
Notice that the x and y values traded places for the function and its inverse.
These functions are reflections of each other about the line y = x
Let’s consider the function and compute some values and graph them.
x f (x)
(2,8)
(8,2)
x f -1(x)
Let’s take the values we got out of the function and put them into the inverse function and plot them
(-8,-2)
(-2,-8)
Yes, so it will have an inverse function
Is this a one-to-one function?
What will “undo” a cube?
A cube root

37. What do we know about the graphs of functions and their inverses?
f and f -1(x) are inverse functions, and f (a) = b for real numbers a and b. Then f -1(b) = a.
If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f -1.
The domain of f is the range of f -1
The range of f is the domain of f -1
If a function is one-to-one, the graph of its inverse f -1(x) is a reflection of the graph of f across the line y = x.

38. What do we know about the graphs of functions and their inverses?
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

39. Think about the domain and range
Is it possible to make a many-to-one functions a one-to-one function, so an inverse does exist?
Think about the domain and range
The domain of f is the range of f -1
The range of f is the domain of f -1
Thus ... we may be required to restrict the domain of f so that f -1 is a function

40. Restricting Domains Not 1-1
Is it possible to make a many-to-one functions a one-to-one function, so an inverse does exist?
Restricting Domains
Not 1-1
Restrict domain to x  0
Now function is 1-1 and has an inverse function
f (x) = x2
You can make a function 1-1 by restricting its domain. Such a function will have an inverse.

41. How do you finding the inverse of a function with a restricted domain?
Example Let
Solution Notice that the domain of f is restricted to [–5,), and its range is [0, ). It is one-to-one and thus has an inverse.
The range of f is the domain of f -1, so its inverse is

42. How do you prove that a function is an inverse of another function?
Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if

43. Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses.
How do you prove that a function is an inverse of another function?
Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses.
Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.
f(g(x))= -3(-1/3x+2)+6
= x-6+6
= x
g(f(x))= -1/3(-3x+6)+2
= x-2+2
= x
** Because f(g(x))=x and g(f(x))=x, they are inverses.

44. Composition of Inverse Functions
How do you prove that a function is an inverse of another function?
Composition of Inverse Functions
Consider
f(3) = 27   and   f -1(27) = 3
Thus, f(f -1(27)) = 27
and f -1(f(3)) = 3
In general   f(f -1(x)) = x   and f -1(f(x)) = x (assuming both f and f -1 are defined for x)

45. Use the composition of functions to show inverses
How do you prove that a function is an inverse of another function?
f and f –1 are inverses if and only if the result of the composition of a function and its inverse (in either order is the original input, x.
Use the composition of functions to show inverses

46. How do you prove that a function is an inverse of another function?
Consider
Demonstrate that these are inverse functions

47. How do you prove that a function is an inverse of another function?
Verify that the functions f and g are inverses of each other.
If we graph (x - 2)2 it is a parabola shifted right 2.
Is this a one-to-one function?
This would not be one-to-one but they restricted the domain and are only taking the function where x is greater than or equal to 2 so we will have a one-to-one function.

48. Verify that the functions f and g are inverses of each other.
How do you prove that a function is an inverse of another function?
Verify that the functions f and g are inverses of each other.
Since both of these = x, if you start with x and apply the functions they “undo” each other and are inverses.

49. Important Facts About Inverses
If f is one-to-one, then f -1 exists.
The domain of f is the range of f -1, and the range of f is the domain of f -1.
If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f -1, so the graphs of f and f -1 are reflections of each other across the line y = x.

50. So geometrically if a function and its inverse are graphed, they are reflections about the line y = x and the x and y values have traded places. The domain of the function is the range of the inverse. The range of the function is the domain of the inverse. Also if we start with an x and put it in the function and put the result in the inverse function, we are back where we started from.
Given two functions, we can then tell if they are inverses of each other if we plug one into the other and it “undoes” the function. Remember subbing one function in the other was the composition function.
So if f and g are inverse functions, their composition would simply give x back. For inverse functions then: