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

DEF: Inverse Relations f(x) and f -1 (x) Switch the x value and the y value 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 x

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 x– 12 =-2y 3 x – 12 = y or f -1 (x) -2 f -1 (x) = - 3x y x

P286 EX: 3 Graph the function defined by y =  x  and its inverse. Is the inverse a function? x  x  function inverse -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 ) y x

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? y x y x 5 5 y x 5 5

Absolute Value 5) y = |x| D: 8) x = |y| D: 10) y=|x|+6 D: x | y Ʀ x | 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? Yes No Yes y x y x 5 5 y x 5 5

Lines 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 x

P286 EX:4 Graph 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 = ½ x + 2 y x