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Functions - Math 16A(011 ha Functions~ Review DEFINITION 2...

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Unformatted text preview: Math 16A [(011 ha Functions~ Review DEFINITION 2 In an equation composed of m’s and y’s, variable y is a function of :E if each admissible x—value has exactly one y—value. NOTE : The graph of a function passes the vertical line test . That is, a vertical line passed through the graph will touch the graph in at most one point. EXAMPLE : Assume that my — 3 = x2 +2y . Then my — 2y : $2 + 3 ——«> (m—2)y:m2+3 ——~> _a:2+3 ._ 23—2 ——> 3/ y is a function of ac. 312—12141; -—> yz—y~2=0 ~*> <y~2><y+n=0 —-—> 3/22 or y=~1 ——> $21 has TWO y—values ——+ y is NOT a function of x . EXAMPLE : If y = x2 + x, then y is a function ofac and we write f(x) = {1:2 + cc; then ayfl—m=4—m2+ta)=4—2=2. h)fflm~lfizQx—1V+(%wd)=4fi—4x+l+2x~1:4fl—flm. DEFINITION 2 Assume that y = f (x) is a function. The domain of function f is the set of all admissible m—values. The range of function f is the set of all corresponding y~values. EXAMPLE : Consider function f(x) : v22: — 6. Then 22: ~ 6 Z 0 ——> 22: 2 6 —-—> a: 2 3 ————> DOMAIN : at 2 3. Since \/233 —- 6 2 0, f (3) = 0, and 22: ~ 6 gets infinitely large as an gets infinitely large, it follows that RANGEzyZO. DEFINITION : A function y = f (at) is one-to-one if each y—value has exactly one x-value. More precisely, a one-to—one function has the property that if f (151) :2 f (272) (y—values are equal), then m1 = mg (x-values are equal). NOTE : The graph of a one-to—one function passes the horizontal line test. That is, a horizontal line passed through the graph will touch the graph in at most one point. EXAMPLE : Consider the function (parabola) y z :02 — 5 . If y z 4, then 4 = x2 - 5 —-—> 3:2 = 9 ——> :5 = 3 or a: = ——3 ———~> y = 4 has TWO m-values ~——> function y is NOT one-tO—one . IL' EXAMPLE : Consider the function f(ac) = + 3 :1: 01182 . Prove that f is one—to- f(331) =f($2) -+ (131 _ 1E2 $1+3—£L'2+3 ——> 1131(5132 + 3) = 332(371 + 3) ——> 561232 + 3331 : mm + 3332 ———> 32:1: 3% —> x1 = x2 ——> function f IS one-to—one . DEFINITION : Assume that y : f(:v) and y 2: 9(33) are functions. The composition of functions f and g is (f 0 9W) = f(9($)) - . . x _ 1 EXAMPLE . ConSIder the functlons f(a:) - 10—2: and 9(53) — 33+8‘ Then (f0 ><->~f< < >>—f 1 g m — gm w $+8 1 10—1-8 1 10“ <x+8) 1 x+8 ‘$+8 10~< 1 > $+8 x+8 1 10(ac+8) —— 1 1 10x+ 79' DEFINITION : The inverse function of function y :2 f (x) is the function y :— f‘1(a:) for which f(f”1($)) = x and f”1(f($)) = $~ FACT : If y = f (:13) is a one—to-one function, then f has an inverse function. SEE INVERSE FUNCTION HANDOUT. W I The function “1”) = :c : 3 is one-to-one. Find its inverse : y 2 :1: i 3 “-> (Switch variables.) a; 2 fig (Solve for y.) $(y + 3) = y ___) my + 3x = y __> :63; “ y = —3:c ——-> Z’J(~T “ 1) = —3w ——> y = 3:33: ———> inverse function is f‘1(x) = 13—1-1; ...
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