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Assignment, pencil, red pen, highlighter, textbook, GP notebook

Determine and graph inverses of functions, practice writing inverse functions, understand composite functions, and identify function properties.

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Assignment, pencil, red pen, highlighter, textbook, GP notebook

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  1. Assignment, pencil, red pen, highlighter, textbook, GP notebook U6D4 Have out: Bellwork: Complete #1 and #2 on the “Graphing Inverses Practice Worksheet”

  2. y 8 x 8 –8 –8 Determine the inverse for each function below. Then graph both functions and the line of reflection. y = x 1) f(x) = 4x – 8 f(x)  multiply by 4 f–1(x)  subtract 8 f–1(x) =  add 8  divide by 4

  3. y 8 x 8 –8 –8 Determine the inverse for each function below. Then graph both functions and the line of reflection. y = x 2) f(x) = x3 – 2 f(x) locator: (0, –2)  cube  subtract 2 f–1(x) f–1(x) =  add 2  cube root To quickly graph the inverse, just switch the x– and y–values.

  4. Let’s take a look at #3 on today’s packet.

  5. y 8 x 8 –8 –8 Determine the inverse for each function below. Then graph both functions and the line of symmetry. y = x 3) f(x) = 2x2 (for domain ) f(x) locator: (0, 0)  square  multiply by 2 f–1(x) Finish the rest of the worksheet by tomorrow. f–1(x) =  divide by 2  square root To quickly graph the inverse, just switch the x– and y–values.

  6. Yesterday, we learned that if we wanted to graph f –1(x) given f(x), then we just had to switch the x and y values. For instance, then This method of switching x’s and y’s can be very useful when we try to write the equation for the inverse of f(x), especially with more complicated functions. For example, how can we quickly find the inverse for f(x)?

  7. Writing Inverse Functions Add to your notes: Steps: Example: 1) Switch the x and y. –5 –5 2) Solve for y. ( )2 2( ) +1 +1

  8. Practice: Write the inverse for each function. 1) 2)

  9. Copy onto your worksheet: Composite Functions input squares The function f(x) _________ all the inputs, while the function g(x) _________ all the inputs. doubles g(x) = 2x f(x) = x2 output What happens if a number goes through both functions? First go through f(x), then go through g(x). Start with x = –1. g(f(–1)) = g(1) (–1)2 f(–1) = g(f(–1)) mean that we take what f(–1) equals and substitute it into g(x). = 2(1) = 1 = 2

  10. Now go through g(x) first, then go through f(x). Start with x = –1 again. g(x) = 2x f(x) = x2 g(–1) = 2(–1) f(–2) f(g(–1)) = = –2 = (–2)2 = 4 order matters Since f(g(–1)) ≠ g(f(–1)), then ______________.

  11. Practice: 1. Given f(x) = 2x + 3 and g(x) = (x – 2)2, find a) f(g(–2)) If you are getting lost in the notation, then follow this example where we do each part separately. g(–2) = (–2 – 2)2 = (–4)2 Solve the problem from the inside out. = 16 Evaluate g(–2) first. f(16) = 2(16) + 3 Evaluate f(16) next. = 32 + 3 = 35 f(g(–2)) = 35 b) g(f(–2)) (–1 – 2)2 g(–1) = Evaluate f(–2) first. f(–2) = 2(–2) + 3 = (–3)2 = –4 + 3 Evaluate g(–1) next. = 9 = –1 g(f(–2)) = 9

  12. Practice: 2. Given and g(x) = x2 + 1, find: a) f(g(5)) b) g(f(5)) f(5) = (5)2 + 1 g(5) = 25 + 1 = 2 = 26 f(26) g(2) = (2)2 + 1 = 4 + 1 = 5 = 5 f(g(5)) = 5 g(f(5)) = 5

  13. Practice: 3. Given f(x) = 2x + 3 and g(x) = (x – 2)2, find a) f(g(x)) b) g(f(x)) g(x) = (x – 2)2 f(x) = 2x + 3 = (2x + 3 – 2)2 g(2x + 3) = f((x – 2)2) = = 2(x – 2)2 + 3 = (2x + 1)2 = 2(x2 – 4x + 4) + 3 = (2x + 1)(2x + 1) = 2x2 – 8x + 8 + 3 = 4x2 + 4x + 1 = 2x2 – 8x + 11 f(g(x)) = 2x2 – 8x + 11 g(f(x)) = 4x2 + 4x + 1

  14. = g( ) Practice: 4. Given and g(x) = x2 + 1, find: c) f(g(x)) d) g(f(x)) = f(x2 + 1) = x – 1 + 1 = x = x f(g(x)) = x g(f(x)) = x Take note that in some textbooks g(f(x)) is written as (g ○ f)(x).

  15. Practice #3: Given f(x) = x + 3, , and h(x) = x2, find: b) (h ○ f)(–4) a) (g ○ f)(9) c) (f ○ h)(5) d) (f ○ g)(x) 9 + 3 f(–4) = –4 + 3 h(5) = f(9) = (5)2 = –1 = 12 = 25 g(12) = (25) + 3 h(–1) = (–1)2 f(25) = = 28 = 1 = 6 – 5 (h ○ f)(–4) = 1 (f ○ h)(5)= 28 = 1 (g ○ f)(9) = 1

  16. Finish today's assignment: Worksheets, CC 32, 35 - 38, 42

  17. take out the packet from yesterday. Let’s finish page 3.

  18. y 5 x 5 –5 –5 3. Sketch y = x2 and y = x. (–∞, ∞) Domain: _________ [0, ∞) Range: __________ y = x f(x) Sketch a graph of the inverse. (2, 4) (–2, 4) Be sure to switch the x– and y–values. (4, 2) Is the inverse a function? (–1, 1) (1, 1) No, it fails the vertical line test. (0, 0) (1, –1) (4, –2)

  19. y 5 x 5 –5 –5 restrict 4. In order for f–1(x) to exist, we must ________ the domain of f(x) = x2. We are just graphing the positive half of the parabola. f(x) = x2 [0, ∞) Domain: _________ f(x) y = x [0, ∞) Range: __________ f–1(x) = f–1(x) [0, ∞) Domain: ___________ [0, ∞) Range: __________

  20. Now go through g(x) first, then go through f(x). Start with x = –1 again. g(x) = 2x f(x) = x2 g(–1) = 2(–1) f(–2) f(g(–1)) = = –2 = (–2)2 = 4 order matters Since f(g(–1)) ≠ g(f(–1)), then ______________. Find the general rule for: For what values does f(g(x)) = g(f(x))? f(g(x)) = f(2x) g(f(x)) = g(x2) 4x2 = 2x2 –2x2 –2x2 They equal when x = 0. = (2x)2 = 2(x2) 2x2 = 0 = 4x2 = 2x2 x2 = 0 x = 0

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