1. Combinations of Functions & Inverse Functions Obj: Be able to work with combinations/compositions of functions. Be able to find inverse functions. TS: Making decisions after reflections and review
Warm-Up:
Given f(x) = 2x + 3 and g(x) = x2 – 4, find…
f(2) + g(2)
g(3) – f(3)

2. Arithmetic Combinations
Where the domain is the real numbers that both f and g’s domains have in common.
For f/g also g(x) ≠ 0.

3. Examples Find each of the below combinations given
(f∙g)(2) ) (f/g)(x) & its domain
(f – g)(2)

4. Arithmetic Compositions (f○g)(x) = f(g(x))
Given the below find each of the following.
(f ○ g)(2) ) g(f(-1))

5. Arithmetic Compositions (f○g)(x) = f(g(x))
Given the below find each of the following.
3) f(g(x)) & its domain

6. Find f(g(x)) and g(f(x))

7. Definition of the Inverse of a Function
Let f and g be two functions where f(g(x)) = x for every x in the domain of g, and g(f(x)) = x for every x in the domain of f. Under these conditions, g is the inverse of f and g is denoted f-1.
Thus f(f-1(x))=x and f-1(f(x))=x where the domain of f must equal the range of f-1 and the range of f must equal the domain of f-1.

8. Graphs of Inverses
Two equations are inverses if their graphs are reflections of one another across the line y=x. y f
y=x
f -1 1 x 1

9. Inverse FUNCTIONS
A function f(x) has an inverse function if the graph of f(x) passes the ___________________.
(In other words the relation is ONE-TO-ONE: For each y there is exactly one x)
Circle the functions that are one-to-one (aka have inverse functions) y y y y x x x x y x

10. Examples:
Determine if the two functions f and g are inverses.
1) and

11. Finding an Inverse
Verify that the function is one-to-one thus has an inverse function using the Horizontal Line Test.
Switch x & y.
Solve for y.
Make sure to use proper inverse notation for y for your final answer. (Ex: f-1(x), not y)
To check your answer:
Verify they are inverses by testing to see if
f(f-1(x)) = f-1(f(x)) = x

12. Find the inverse function if there is one, if there is not one, restrict the domain to make it one-to-one then find the inverse function .
f(x) = -.5x + 3

13. Find the inverse function if there is one, if there is not one, restrict the domain to make it one-to-one then find the inverse function .
2) f(x) = x2 – 4