1. 6-7 Inverse Relations and Functions
Find the inverse of a relation or function by relating domain and range.

2. Inverse Relation
If a relation pairs elements a in the domain with elements b in the range, its inverse pairs b with a.
If (a, b) is an ordered pair in a relation, (b, a) is an ordered pair of the relation’s inverse
If both a relation and its inverse are functions, they are inverse functions.

3. Finding the Inverse Create the table which shows the inverse of s
Sketch the graph of s
Now sketch the graph of the inverse
Notice that the inverse is reflected
across the line y = x

4. Finding an Equation for the Inverse
What is the equation of the inverse of 𝑦= 𝑥 2 −1
Since the inverse is found by switching the domain and range, switch x and y
𝑥= 𝑦 2 −1
Now solve for y
𝑦=± 𝑥+1
You try: find the inverse of 𝑦=2𝑥+8
𝑦= 1 2 𝑥−4

5. Graphing Inverses
The graph of the function 𝑦= 𝑥 2 −1 is a parabola with vertex (0, -1)
The inverse relation is the reflection of that parabola across the line y = x.

6. Finding an Inverse Function
Denoted 𝑓 −1 and read as “inverse of f” or “f inverse”
What is 𝑓 −1 if 𝑓 𝑥 = 𝑥−2
Write as 𝑦= 𝑥−2
Now switch and solve for y: 𝑥= 𝑦−2
𝑥 2 =𝑦−2
𝑦= 𝑥 2 +2
𝑓 −1 𝑥 = 𝑥 2 +2

7. Domain and Range
The domain of the inverse is the same as the range of the original relation.
Ex: 𝑓 𝑥 = 𝑥−2 and 𝑓 −1 𝑥 = 𝑥 2 +2
The domain of 𝑓 −1 is the same as the range for f.
So the domain is all real numbers x ≥0 or [0,∞)
The range of 𝑓 −1 is the same as the domain for f
So the range is all real numbers 𝑦≥2 or [2,∞)
If each x-value maps to exactly one y-value and it is also true that each y-value maps to exactly one x-value, then the function is called a one-to-one function.

8. Composition of Inverse Functions
If 𝑓 𝑥 = 1 𝑥−1
Find 𝑓∘ 𝑓 −1 1
Start by finding 𝑓 −1 (𝑥): 𝑓 −1 𝑥 = 1 𝑥 +1
Now evaluate 𝑓 −1 (1): 𝑓 −1 1 = =2
Find 𝑓 2 = 1 2−1 =1
Now find ( 𝑓 −1 ∘𝑓)(1)
Since 1 is not in the domain of f, we cannot find 𝑓(1) and 𝑓 −1 ∘𝑓 1 does not exist.

9. Assignment
Odds p.410 #11-25, 31-35,39,41,49