Composition of Functions Definition: ForMath functions 140 f andLectureg, define 6f ◦ g, the compositionWriteof f andeachg functionby, below as a composition Study Practice Exam 1 and the recommended(f ◦ exercises.g)(x) = f (g(x)) f (g(x)) of two simpler functions, Functions can be added and multiplied just like numbers. an outer function f and an inner function g. DEFINITION. For functions f, g, define f +g, f - g, f.g, f /g Find the inner function first. by Dom(f ◦ g) = fx 2 dom(g)jg(x) 2 dom(f)g `Write x2 26 as a composition fgx. (f + g)(x) = f (x) +g(x), Note: (f +g)(x) is not (f +g).(x) Example: Suppose f(x) = x − 2 and g(x) = x2. x2 26 inner function gxx2 2 (f - g)(x)= f (x)- g(x), The first is function application. (fg)(x) = f (x)g(x), The second is multiplication. „ outer function fx does what remains (a) Find (f ◦ g) and (g ◦ f). 6 (f /g)(x) = f (x)/g(x). x to be done. t fxx6. `If f (x) = x- 2, g(x) = 6, then 2 2 (f ◦ g)(x) = f(g(x)) = f(x ) = x − 2 check: fgx f x2 2x2 26. (f + g)(x) = f (x) + g(x) = (x- 2) + (6) = x + 4, 2 2 1 (f-g)(x()g =◦ f ()(xx) )- = g(gx()f (=x ))(x =- 2)g(x -− (6)2) = = x ( x- −8,2) = x − 4x + 4`Write 4 x 3 as a composition f gx. 1 1 (fg)(***Note:x) = f (x)g((fx)◦ =g )(6=x- (2)(6)g ◦ f) = 6x-12, 4 x 3 inner function gx x 1 1 (f /g)(x) = f (x)/g(x) = (x-2)/6 = 6 x 3 . „ outer function f x does what remains DEFINITION(b) Find. For (f functions◦ g)(2) and f and (g ◦ gf,)(2). define fog, the 4x 3 to be done. t composition of f and g, by f x4x 3. (f ◦ g)(2) = f(g(2)) = f(22) = f(4) = 4 − 2 = 2 1 1 (fog)(x) = f (g(x)) check: fgx f x 4 x 3. 2 Apply g to(g ◦x.f Get)(2) g =(xg).(f Apply(2)) = gf(2 to− g2)(x =). gGet(0) f = (g 0(x=)). 0 `Write x 1 as a composition. f is the outer function; g is the inner function. x 1 Example: For f(x) = 3x + 4 and 2g(x) = 5, find (f ◦ g) and (g ◦ f). `Suppose f (x) = x-2 and g(x) = x . „ gxx 1 (a) Find f(ofg◦ andg)(x g)o =f. f(g(x)) = f(5) = 3(5) + 4 = 19 2 2 x f x x (fog)(x) = f(g(x)) = f (x ) = x -2. 2 2 check: f gx f x 1 x 1 . (gof)(x) (=g g◦(f(x)())x )= =g(xg-2)(3x =+ (x 4)-2) = = 5 x - 4x + 4. Note that f og = gof. For composition, order matters. `Write x4 x2 1 as a composition. (b) Example:Find (fog)(2).For f and g below, note that x4 x2 1 x22 x21. 2 2 Since (fofg(−)(x3)) = = x 1,-2,f ((f−og1))(2) = = 2, 2f-2(2) = =4-2 3, =f 2.(4) = 2, and 2 g(−2) = −1, g(−1) = −2, g(1) = −1, g(2) = 2. gxx (b’) Find (fog)(2) and (gof )(2) directly without (fog)(x), 2 Find: f xx x 1 (gof )(x). 2 4 2 (fog)(2) = f(g(2)) = f(22) = f(4) = 4-2 = 2. check: f gx fx x x 1. 2 (gof)(2)( =f g(f(2))◦ g)(2) = = g(2-2)f(g(2)) = g(0) = f =(2) 0 == 0. 3, `Write x /1 x as a composition of 2 functions. 2 `Write as a composition of 3 functions. `If h(x) = c, then h(8) = c, h(y) = c, h(x -1) = c, h(g(x)) = c. 1/1 |x| (g ◦ f)(2) = g(f(2)) = g(3) = undefined, Ans hfgx, gx|x|, fx1 x, hx1/x `For f (x) = 3x+4, g(x) = 5, find fog and gof. f(g(x)) = f(5) = 3.5+4 = 19. g(f(x)) =( fg(3◦ xf+4))(− =1) 5. = f(f(−1)) = f(2) = 3 DEFINITION. id(x) = x is called the identity function. id(x)=x f(x) g(x) 2 2 Hence id(5) = 5, id(y) = y, id(x -1) = x -1, ... For f and g above, note that THEOREM. For any function f (x), foid = f and idof = f. f(-3) = 1, f(-1) = 2, f(2) = 3, f(4) = 2, and PROOF. ( foid)(x) = f (id(x)) = f (x). g(-2) = -1, g(-1) = -2, g(1) = -1, g(2) = 2. (idof )(x) = id(f (x)) = f (x). Find 0 is the identity for addition, since f + 0 = 0 + f = f . (fog)(2) = f(g(2)) = f(2) = 3. 1 (gof)(2) = g(f(2)) = g(3) = undef. 1 is the identity for muliplication, f .1 = 1. f = f . (fof)(-1) = f(f(-1)) = f(2) = 3. id(x) is the identity for composition, since 1 1 1 1 x `fxx x , ffx x x 1/x x x x x21 foid = idof = f . 1 Example: f(x) = x + x . Find (f ◦ f). 1 1 1 x (f ◦ f)(x) = f(f(x)) = (x + x ) + ( 1 ) = x + x + x2+1 x+ x Next we want to write a function as a composition of 2 simpler functions. Example: Write (x2 + 2)6 as a composition f(g(x)). (x2 + 2)6 has an inner function g(x) = x2 + 2. Then the outer function f(x) does what remains to be done: f(x) = x6. Check: f(g(x)) = f(x2 + 2) = (x2 + 2)6. 1 Example: Write 4 x + 3 as a composition f(g(x)). 1 1 4( x ) + 3 as inner function g(x) = x . Then the outer function f(x) does what remains to be done: f(x) = 4x + 3. 1 1 Check: f(g(x)) = f( x ) = 4( x ) + 3. p Example: Write x + 1 as a composition f(g(x)). p x + 1 has inner function g(x) = x + 1 p So f(x) = x. p Check: f(g(x)) = f(x + 1) = x + 1. Example: Write x4 + x2 + 1 as a composition. x4 + x2 + 1 = (x2)2 + x2 + 1. ) g(x) = x2 ) f(x) = x2 + x + 1. Check: f(g(x)) = f(x2) = (x2)2 + x2 + 2 = x4 + x2 + 1. 1 Example: Write 1+jxj as the composition of 3 functions h(f(g(x))). 1 1+jxj has inner function g(x) = jxj 1 (1+x) has inner function f(x) = 1 + x 1 Thus h(x) = x . 1 Check: h(f(g(x))) = h(f(jxj)) = h(1 + jxj) = 1+jxj 2 -1 `If f (x) = x+3 then f (-1x) = x-3. Math 140 Lecture 7 `If f (x) = x+3 then f (x) = x-3. Math 140 Lecture 7 -1 If g(x) = x/2 then g (-1x) = 2x. Inverse functions If g(x) = x/2 then g (x) = 2x. Inverse functions -1 2 -1 If h(x) = x then h (-1x) = x for2 x > 0. DEFINITION. f ,-1 the inverse of f , is the function, if any, If h(x) = x then h (x) = x for x > 0. DEFINITION. f , the inverse of f , is the function, if any, -1 Note how the graph of f is related to the graph of f .-1 such that, such that, Note how the graph of f is related to the graph of f . -1 -1 f ( f (-1x)) = x when f (-1x) is defined and -1 f ( f (x)) = x when f (x) is defined and g -1 -1 g h f (-1 f (x)) = x when f (x) is defined. h f ( f (x)) = x when f (x) is defined. g -1-1 g -1 -1 f f -1 This says that f and f -1undo each other: f f h This says that f and f undo each other: h -1 f -1undoes what f (x) does and gives you back x. f undoes what f (x) does and gives you back x. -1 -1 -1 `f(x) =2x, f (-1x)=½x. Verify: f(f (-1x))=x & f ( -1f (x))=x `f (x) = 2x, f (x) = ½x. Verify: f (f (x)) = x & f (f (x)) = x -1 -1 `g(x) = x +3, g (-1x) = x-3. By the Theorem, y = f (-1x) iff f (y) = x. Thus the graph of `g(x) = x +3, g (x) = x-3. By the Theorem, y = f (x) iff f (y) = x. Thus the graph of -1 1 3 -1 `h(x) = 2x +3, h (-1x) = (x-3)/2 = x1 .3 y = f (-1x) is the graph of f (y) = x which is just the graph `h(x) = 2x +3, h (x) = (x-3)/2 =2 x 2 . y = f (x) is the graph of f (y) = x which is just the graph -1 2 2 Verify that h (-1h(x)) = x. of f (x) = y with x and y interchanged. Interchanging x Verify that h (h(x)) = x. of f (x) = y with x and y interchanged. Interchanging x To undo a sequence of operations, you must undo them in and y reflects the plane around the major diagonal y = x. To undo a sequence of operations, you must undo them in and y reflects the plane around the major diagonal y = x.