Inverse Functions

1. Inverse Functions

2. End In Mind • Prove that and are inverses of each other • Complete warm up individually and then compare to a neighbor

3. Vocabulary • Function: a relation in which each input x has exactly 1 output y • Inverse of a Function: The inverse function is a function that undoes another function: If an input x in the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. • One to One: A function is one to one if every output y, has exactly 1 input x. • Horizontal Line Test - A function f is one to one if and only if each horizontal line intersects the graph at most once. • Composition of Functions: is the process of combining two functions where one function is performed first and the result of which is substituted in place of each x in the other function. The composition of functions f and g is written as f o g. [f o g](x) = f[g(x)]

4. Inverse Functions • An inverse function undoes what the function does f(x) range domain Can you mentally determine the inverse of the functions?

5. Solving for an inverse algebraically Finding the inverse of a function Problem: • Replace f(x) with a y • Switch the x and y • Solve for y • Replace y with f-1(x)

6. Find each inverse then check your solution with a friend

7. Finding compositions • To find a composition of 2 functions, substitute one function for the other function: Example: f(x)= 3x-8 and g(x) = x2+1 • To find f(g(x))=f○g(x) substitute the g(x) function for the f(x) function f(g(x))=f○g(x) = 3(x2+1)-8 =3x2+3-8 =3x2-5 • To find g(f(x))=g○f(x), substitute the f(x) function for the g(x) function: • g(f(x))=g○f(x) = (3x-8)2+1 • =(3x-8)(3x-8)+1 • =9x2-48x+64+1 • =9x2-48x+65

8. Verifying inverses an application of compositions To verify that two functions are inverses then, Using our earlier problem, Verify that and are inverses of each other.

9. End In Mind • Prove that and are inverses of each other

10. End In Mind • Determine the inverse values of the function from the table:

11. Inverses Graphically • Graphing inverse functions The graph of the inverse of f is the reflection of f over the line y=x

12. Horizontal Line Test • Existence of an Inverse function a function f has an inverse function if and only if the function is one to one. • One to One a function f is one to one if for every y there is exactly one x value • Horizontal line test

13. Determine whether the function is invertible. If it is, find it’s inverse a. {(4,3),(2,-1),(5,6)} b. {(9,0),(8,1)(,4,0)} c. d. Yes. {(3,4),(-1,2),(6,5)} Not invertible. Since 2 y values are the same. Not invertible since all y values are the same.

14. End In Mind • Determine the inverse values of the function from the table:

15. Use y=x2+5 End in Mind: limit the domain so that the inverse is a function Is the relation a function? Graph the function. Does the inverse exist? How could you limit the domain so that the function will have an inverse? Graph the inverse with the restricted domain. How can you verify that the graph of the inverse exits?

16. Ticket Out http://www.regentsprep.org/Regents/math/algtrig/ATP8/indexATP8.htm GO to the above website for further explanations. You must do the practice problems. Each problem will tell you if you are right or wrong. If you need help, click the explanation button