1. Watch this!! http://courseware.cemc.uwaterloo.ca/8/122/assignments/113
1.5 The Inverse Function
Watch this!!

2. Inverse undoing an operation examples operation: tying shoe laces
inverse: untying shoe laces
Converting between oC and oF

3. Inverse
undoing an operation

4. A Function & Its Inverse Function
???

5. A Function & Its Inverse Function

6. A Function & Its Inverse Function
Notice anything interesting?

7. A Function & Its Inverse Function
Notice anything
interesting?
input of f(x) = output of g(x)
output of f(x) = input of g(x)
What does that mean?

8. A Function & Its Inverse Function
Notice anything
interesting?
domain of f(x) is the range of g(x)
range of f(x) is the domain of g(x)

9. Inverse
undoing an operation

10. Inverse Function
The inverse of a function is a relation which undoes the work of the function.
The inverse of a function may or may not be a function (it may or may not pass the vertical-line test).
When the inverse of a function, f(x), is also a function, it is denoted as f−1(x).
To find the inverse of a function (or relation), we interchange (or switch) x and y.

11. Determine the Inverse of a Function Reversing the Operations
What does this function tell you to do? (BEDMAS)
multiply by (-5)
add 2
What is the reverse? How do you undo the above?
subtract 2
divide by (-5)

12. Determine the Inverse of a Function Switching the Variables (x  y)
Switch the variables.
Isolate (solve) for y.

13. EXAMPLE 1
Algebraically, determine the inverse of f(x). You can reverse the order of operations or you can switch the variables and solve for y.
Graph f(x) and its inverse on Desmos.
State the domain and range of f(x) and of its inverse.
Is the inverse of f(x) a function?

14. EXAMPLE 1 x  y

15. EXAMPLE 1 Reverse Order subtract 5 2 divided by answer subtract 4
add 4
add 5

16. EXAMPLE 1 Reverse Order

17. INVESTIGATION TIME 
Complete 1.5 Investigation – Inverse of a Function

18. EXAMPLE 2 Determine the inverse. Graph the inverse.
Is the inverse a function?
What is the domain and
range of f(x) and its inverse?

19. EXAMPLE 2 Determine the inverse. Graph the inverse.
Is the inverse a function?
f(x) = { (-3,1), (-2, 1), (-1,1), (0,3), (1, 0), (4, -2)}
f-1(x) = {(1, -3), (1, -2), (1, -1), (3, 0), (0,1), (-2, 4)}
The inverse is not a function. The vertical-line test fails at (1, -3), (1, -2) & (1, -1).

20. EXAMPLE 2 Determine the inverse. Graph the inverse.
Is the inverse a function?
What is the domain and
range of f(x) and its inverse?
f(x)
f-1(x)
domain
{-3,-2,-1,0,1,4}
{-2, 0, 1, 3}
range
{-2, 0, 1, 3 }

21. EXAMPLE 2 Determine the inverse. Graph the inverse.
Is the inverse a function?

22. EXAMPLE 2 Determine the inverse. Graph the inverse.
Is the inverse a function?
Inverse of a linear function is also a linear function.
Find two points on the line g(x) : (0,3) and (5, 0).
Two points on the line g-1(x) will be (3, 0) and (0, 5).
Draw a line through those points.

23. EXAMPLE 3 Given f (x) = { (1 , 3) , (-2 , 1) , (-5 , -1) }
Find f – 1(x).
Graph both functions on the grid.
Is f – 1(x) a function?
How are f (x) and f – 1(x) related?

24. EXAMPLE 3 Given f (x) = { (1 , 3) , (-2 , 1) , (-5 , -1) }
Refer to graph.
f – 1(x) is a function
f – 1(x) is the reflection
image of f (x) in the line y = x

25. Summary - Inverse of a Function
The inverse of a function undoes whatever operations are performed by the function.
To form an inverse from a set of ordered pairs, interchange the x and y coordinates.
Geometrically, an inverse is found by reflecting the function (or relation) about the line y=x.
domain of the inverse = range of function
range of inverse = domain of function
To determine the equation of an inverse algebraically, interchange x and y in the equation of the relation and solve the resulting equation for y.

26. HomeFUN 
pg #1, 2, 3, 4(use Desmos), 5, 6, 7, 11, 12, 13