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Warmup Alg 31 Jan 2012

Warmup Alg 31 Jan 2012. 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. Practice from last class period’s assignment. Section 6.4: Inverses of Functions.

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Warmup Alg 31 Jan 2012

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  1. WarmupAlg31 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) = 5x2 – 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( ) • 4x4x f(g(x)) = 5(16x2 ) – 2( 4x ) f(g(x)) = 80x2 – 8x

  7. Composition of Functions 2 If f(x) = 5x2 – 2x and g(x) = 4x • Then g(f(x)) is: • g(f(x)) = 4( ) • f(x) g(f(x)) = 4( ) • 5x2 – 2x g(f(x)) = 20x2 – 8x

  8. Composition of functions 3 • If f(x)=2x and g(x) = 2x2+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) = 2x2+2 and h(x)= -4x + 3 • Find h(g(2)) h(g(2)) = -4( ) + 3 • g(2) • 2(2)2 +2 h(g(2)) = -4( ) + 3 h(g(2)) = -4( )+ 3 • 2(4)+2 h(g(2)) = -4( 10 )+ 3 = -37

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

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

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

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

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

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

  16. Checking Inverses #2 • Can you show that • y = 2x + 3 and • y = ½ x – 3/2are 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”

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

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