1. Form a Composite Functions and its Domain

2. Composite Function
Given two functions f and g, the composite function, denoted by f ◦ g (read as f composed with g) is defined by (f ◦ g)(x) = f(g(x)) The domain of f ◦ g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

3. Evaluating a Composite Function
Example 1
Suppose that f(x) = 2x3-3 and g(x) = 4x
Find
(f ◦ g)(1)=
(g ◦ f)(1) =
(f ◦ f)(-2) =
(g ◦ g) (-1) =

4. Finding a Composite Function
Example 2
Suppose that f(x)= x2+3x-1 and g(x) = 2x+3
Find
f ◦ g and state the domain of each composite function

5. Finding a Composite Function
Example 3
f(x)= x2+3x-1 and g(x) = 2x+3
g ◦ f and state the domain of each composite function

6. Finding the Domain of f ◦ g
Find the domain (f ◦g )(x) if f(x) = and g(x) =

7. Solution
Domain of g is {x│x ≠ 1} Therefore 1 has to be excluded from the domain of the composite Domain of f is {x│x≠ -2} This means g(x) can never equal -2 we need to solve for g(x) = -2

8. Finding a Composite Function
Suppose that f(x)= and g(x) = Find f◦ g Find f ◦ f

9. Showing that two Composite Functions are equal
If f(x) = 3x-4 and g(x) = show that (f ◦ g)(x) = (g ◦ f)(x)= x

10. Homework
Page 229 examples 7-45 every third question

11. Inverse Functions Determine the Inverse of a Function
Obtain the Graph of the Inverse Function from the graph of the Function
Find the Inverse Function f-1

12. What is the Inverse?
Something we know: We know that a function can be compared to a machine. We input an x and the machine manipulates it and outputs the value f(x) or the y value. The inverse of f receives as input a number f(x) and manipulates it and outputs the value x.

13. Finding the inverse of a Function
Let the domain of the function represent the people we know and let the range represent their birthdays. Domain Range
Mrs Ireland December 7
Bry’Eisha March 11
Melissa October 7
Allena March 1

14. Finding the inverse of a function
Find the inverse of the following functions: {(-3,9), (-2,4), (-1,1), (0,0), (1,1), (2,4), (3,9)} The inverse of the given function is found by interchanging the entries in each ordered pair Answer: {(9,-3),(4,-2),(1,-1),(0,0),(1,1),(4,2)(9,3) Is the inverse a function? No because there are two outputs for 4, 9 and 1

15. Finding the inverse of a function
Example 2: Find the inverse of the function {(-3,-27),(-2,-8),(-1,-1),(0,0),(1,1),(2,8),(3,27) Answer: {(-27,-3),(-8,-2),(-1,-1),(0,0),(1,1),(8,2),(27,3) Is the inverse a function? Yes! Why?

16. One to one
When the inverse of a function f is itself a function then f is said to be a one to one function. That is f is one to one if for any choice of elements x1 and x2 in the domain of f, with x1 ≠ x2, the corresponding values f(x1) and f(x2) are unequal, f(x1) ≠ f(x2) In words A function is one to one if any two different inputs never correspond to the same output

17. Horizontal Line Test
Something we know: We know to test whether a graph is a function we use the vertical line test Now we have a new test Horizontal Line Test: If every horizontal line intersects the graph of a function f in at most one point, then f is one to one.