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Unformatted text preview: Math 16A
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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 onetoone if each y—value has exactly
one xvalue. More precisely, a oneto—one function has the property that if
f (151) :2 f (272) (y—values are equal), then m1 = mg (xvalues are equal). NOTE : The graph of a oneto—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 mvalues ~——> function y is NOT onetO—one . IL' EXAMPLE : Consider the function f(ac) = + 3
:1: 01182 . Prove that f is one—to f(331) =f($2) + (131 _ 1E2
$1+3—£L'2+3 ——> 1131(5132 + 3) = 332(371 + 3) ——>
561232 + 3331 : mm + 3332 ———>
32:1: 3% —>
x1 = x2 ——> function f IS oneto—one . DEFINITION : Assume that y : f(:v) and y 2: 9(33) are functions. The
composition of functions f and g is (f 0 9W) = f(9($))  . . x _ 1
EXAMPLE . ConSIder the functlons f(a:)  10—2: and 9(53) — 33+8‘
Then
(f0 ><>~f< < >>—f 1
g m — gm w $+8
1
10—18 1
10“ <x+8) 1
x+8 ‘$+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—toone function, then f has an inverse function. SEE INVERSE FUNCTION HANDOUT. W I The function “1”) = :c : 3 is onetoone. 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—11; ...
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 Spring '10
 Kouba
 Calculus

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