1. Composite Function: Combining a function within another function.
Notation:
Function “f” of Function “g” of x
“x to function g and then g(x) into function f”
Example 1 Evaluate Composites of Functions
Recall: (a + b)2 = a2 + 2ab + b2 a) b)

2. Example 2 Composites of a Function Set
g(x)
f(x)
f(g(x)) X Y 3 5 7 9 11 X Y 7 3 5 9 8 11 4 X Y 3 5 7 8 9 4
This means f(g(5))=3

3. Example 2 Composites of a Function Setb)
f(x)
g(x) X Y 7 3 5 9 8 11 4 X Y 5 7 3 9 11
In set form, not every x-value of a composite function is defined

4. Evaluate Composition Functions
f(g(3)) b) g(f(-1))

5. Evaluate Composition Functions
c) d)

6. Inverse Functions and Relations
Inverse Relation:
Relation (function) where you switch the domain and range values
Function  Inverse
Function  Inverse
Domain of the function  Range of Inverse and Range of Function  Domain of Inverse
Inverse Notation:
Inverse Properties: 1]
Input a into function and output b, then inverse function will input b and output a (switch) 2]
Composition of function and inverse or vice versa will always equal x (original input)

7. Steps to Find Inverses One-to-One: [1] Replace f(x) with y
ONE TO ONE: There are no repeated x or y values.
Every x is paired with exactly one unique (DIFFERENT) y-value
[1] Replace f(x) with y
[2] Interchange x and y
[3] Solve for y and replace it with
One-to-One:
A function whose inverse is also a function (horizontal line test)
Function
Inverse
Inverse is not a function

8. Example 1: Inverses of Ordered Pair Relationsb) c)
Are inverses f-1(x) or g-1(x) functions?
f-1(x) not a function because x repeats

9. Inverses of Graphed Relations
FACT: The graphs of inverses are reflections about the line y = x
Find inverse of y = 3x - 2
y = 1/3x + 2/3
x = 3y – 2
x + 2 = 3y
1/3x + 2/3 = y
y= x
y= 3x - 2

10. Example 3 Find an Inverse Functionb)

11. Example 3 Continued c) d)
PART D) Function is not a 1-1. (see example)
So the inverse is 2 different functions:
If you restrict the domain in the original function, then the inverse will become a function. (x > 0 or x < 0)

12. Example 4: Verify two Functions are Inverses
Method 1: Directly solve for inverse and check
Method 2: Composition Property
Yes, Inverses
Yes, Inverses