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11.4 Inverse Relations and Functions

11.4 Inverse Relations and Functions. OBJ:  Find the inverse of a relation  Draw the graph of a function and its inverse  Determine whether the inverse of a function is a function. P 285 EX A = {( 1 ,2), ( 2 ,-3), ( 5 ,2)} Find A -1

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11.4 Inverse Relations and Functions

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  1. 11.4 InverseRelations and Functions OBJ: Find the inverse of a relation Draw the graph of a function and its inverse Determine whether the inverse of a function is a function

  2. P 285 EX A = {(1,2), (2,-3), (5,2)} Find A-1 A-1 = {(2, 1), (-3, 2), (2, 5 )} EX1A ={(-3,-2),(-1,2),(3,5)} NOTE: When both a relation and its inverse are functions, they are Called inverse functions. FIND: A-1 A-1= {(-2,-3),(2,-1),(5,3)} Is A-1 a function? Yes y 5 x -5 -5 5 DEF: InverseRelations f(x) and f-1(x)Switch the xvalueand the y value

  3. y 5 x -5 -5 5 NOTE: When the inverse of a given function “f” is a function, f –1 is used to denote it. If a function is defined by an equation, the equation of the inverse is obtained by interchanging x and y in the original equation. P 285 EX: 2 A function “f” is defined by the equation y = -2/3 x + 4. Find an equation for f –1(x) x = -2/3 y + 4 (x = -2/3 y + 4)3 3 x = -2 y + 12 3 x– 12 =-2y 3 x – 12 = y or f-1(x) -2 f-1(x)= - 3x+ 6 2

  4. y 5 x -5 -5 5 P286EX: 3 Graph the function defined by y = xand its inverse. Is the inverse a function? x xfunctioninverse -2 _______________________________ 2 ( -2, 2 ) ( 2, -2 ) -1 ______________________________ 1 ( -1, 1 ) ( 1, -1 ) 0 _________________________ 0 ( 0, 0 ) ( 0, 0 ) 1 ______________________________ 1 ( 1, 1 ) ( 1, 1 ) 2 ______________________________ 2 ( 2, 2 ) ( 2, 2 )

  5. y y y 5 5 5 x x x -5 -5 -5 -5 -5 -5 5 5 5 Absolute Value 5) y = |x| D: 8) x = |y| D: 10) y=|x|+6 D: x | y x | y x | y (-1, ) R: (1, ) R: (-1, ) R: (0, ) (0, ) (0, ) (1, ) F ? (1, ) F ? (1 , ) F?

  6. y y y 5 5 5 x x x -5 -5 -5 -5 -5 -5 5 5 5 Absolute Value 5) y = |x| D: 8) x = |y| D: 10) y=|x|+6 D: x | y Ʀx | y [0, ∞)x | y Ʀ (-1, 1) R: (1, -1) R: (-1, 7) R: (0, 0) [0, ∞) (0, 0 ) Ʀ (0, 6) [6, ∞) (1, 1) F ? (1, 1 ) F ? (1 , 7) F? YesNo Yes

  7. Ax + By = C y = mx + b 9) 5x – 2 y = 10 (0, -5) (2, 0) 2y = -5x + 10 y = 5/2 x – 5 Standard Form Slope-Intercept Form Standard Form Cover-up method Slope-Intercept Form y 5 x -5 -5 5 Lines

  8. y 5 x -5 -5 5 P286 EX:4Graph the function defined by 4x–2y =8 and its inverse in the same coordinate plane. Is its inverse a function? 4x – 2y = 8 Use cover-up (0, ) ( ,0) (0,-4) (2,0) 4y – 2x = 8 Use cover-up (0, ) ( ,0) (0,2) (-4,0) Or get y [f-1(x)] by itself 4y = 2x + 8 f-1(x) or y = ½x+ 2

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