1. Warmup Alg 31 Jan 2012

2. Agenda Don't forget about resources on mrwaddell.net Section 6.4: Inverses of functions Using “Composition” to prove inverse Find the inverse of a function or relation

3. Practice from last class period’s assignment.

4. Section 6.4: Inverses of Functions

5. Vocabulary Domain Range Inverse Composition The x values of the points The y values of the points A Function “flipped” Putting one function inside another

6. Composition of Functions If f(x) = 5x 2 – 2x and g(x) = 4x Then f(g(x)) is: f(g(x)) = 5( ) 2 – 2( ) g(x) g(x) f(g(x)) = 5( ) 2 – 2( ) 4x 4x f(g(x)) = 5(16x 2 ) – 2( 4x ) f(g(x)) = 80x 2 – 8x

7. Composition of Functions 2 If f(x) = 5x 2 – 2x and g(x) = 4x Then g(f(x)) is: g(f(x)) = 4( ) f(x) g(f(x)) = 4( ) 5x 2 – 2x g(f(x)) = 20x 2 – 8x

8. Composition of functions 3 If f(x)=2x and g(x) = 2x 2 +2 and h(x)= -4x + 3 Find g(h(2)) g(h(2)) = 2( ) 2 + 2 h(2) g(h(2)) = 2( ) 2 + 2 -4(2) + 3 g(h(2)) = 2( ) 2 + 2 -8 + 3 g(h(2)) = 2( -5 ) 2 + 2 = 52

9. Composition of functions 3 If f(x)=2x and g(x) = 2x 2 +2 and h(x)= -4x + 3 Find h(g(2)) h(g(2)) = -4( ) + 3 g(2) h(g(2)) = -4( ) + 3 2(2) 2 +2 h(g(2)) = -4( ) + 3 2(4)+2 h(g(2)) = -4( 10 )+ 3 = -37

10. Inverse #1 Find the inverse of each relation, state whether the inverse is also a function. 1. {(1, 2), (2, 4), (3, 6), (4, 8)} a. Switch the X and Y values around {(2, 1), (4, 2), (6, 3), (8, 4)} b. Are there any repeats in the NEW X positions? Then it IS a function! Original Inverse

11. What we are doing The inverse “flips” the picture over!

12. Inverse #1 Find the inverse of each relation, state whether the inverse is also a function. 1. {(1, 3), (4, 2), (6, 9), (6, 5)} Is this a Function? Inverse = {(3, 1), (2, 4), (9, 6), (5, 6)} No Yes! 1. {(6, 7), (8, 3), (5, 3), (9, 8)} Inverse = {(7, 6), (3, 8), (3, 5), (8, 9)} Yes No

13. Inverse # 2 Find an equation for the inverse of: y = 2x + 3 First, switch the x and y Second, solve for y. x = 2y + 3 -2y -2y +x = + 3 -x -x -2y = -x + 3 -2 -2 -2 y = ½ x – 3/2 That’s all there is to it.

14. Inverse #2 Now you try. Find the inverse of: y = - ½x + 3 The inverse is: y = -2x + 6 Prove it! Here’s how.

15. Verifying an inverse is true. f(x) = - ½x + 3 and the inverse is: g(x) = -2x + 6 f(g(x)) = -½( ) + 3 f(g(x)) = -½( -2x + 6 ) + 3 f(g(x)) = x - 3 + 3 f(g(x)) = x g(f(x)) = -2( ) + 6 g(f(x)) = -2(-½ x + 3) + 6 g(f(x)) = x - 6 + 6 g(f(x)) = x Do (f◦g) and (g◦ f) and if they both equal “x” then they are inverses!

16. Non-linear inverse functions The dashed line is the equation: y = x Notice the symmetry in the red and blue graphs!

17. Non-linear inverses The dashed line is the equation: y = x Notice the symmetry in the red and blue graphs!

18. Checking Inverses #2 Can you show that y = 2x + 3 and y = ½ x – 3/2 are inverses of each other? Do f(g(x)) and g(f(x)) and if they both equal “x” then they are inverses! Hint: Call the first one “f(x)” and the second one “g(x)” and lose the “y’s”

19. Assignment Section 6.4: 6-11, 15-20, 42-43 Do All, and pick 1 from each group to write complete explanation.