Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g: D, we define the lgebic opetions of functions. ( f± g)( ) = f( ) ± g ( ). 2. ( f g)( ) = f( ) g ( ). 3. f ( ) = g f( ) g ( ), if g ( ) 0. Let f : D nd g: D be functions nd f( D) D, we define the composition of f with g, denoted g f by ( g f )( ) = g( f( ) ). f( ) g( f( )) D f D g The most common method fo visulizing function is its gph. If f : D is function, then its gph is the set of ode pis {(, f( ) ) D} Symmety A function f : D is sid to be odd if f( ) = f( ) fo ll D. A function f : D is sid to be even if f( ) = f( ) fo ll D.
Note tht The gph of n odd function is symmetic bout the oigin, nd n even function is symmetic bout the y is. The Eponentil Functions Given positive numbe, we ll know wht mens when is tionl numbe. Futhemoe, if nd b e positive numbes, nd s e tionl numbes, then we hve the Lws of eponents: s s. + s = 2. = s 3. ( ) s s = 4. ( ) b = b Hee is the gph of ( ) {,2 = 2,.9,..9,2} If we plot moe points, we would get smooth line pssing though ll the points of the fom (,2 ), whee is tionl numbe. This suggests tht we should be ble to define 2 fo ll el numbe. Question : Wht is 2 2? Actully, we should nswe wht is 2? By the bisecting method, we hve.4 2.5.4 < 2 <.5 2 < 2 < 2,.4 2.425.4 < 2 <.425 2 < 2 < 2,.4 2.45.4 < 2 <.45 2 < 2 < 2,.425 2.425.425 < 2 <.425 2 < 2 < 2.
Finlly, we cn get So, we cn define 2 f( ) = 2 2.445625 2 2. whee is ny el numbe. Hee is the gph of the function The gph of membes of eponentil functions vlues of bse. y = e shown below fo vious Note tht We cn see tht ll the gphs of eponentil functions pss though the sme point (0,) becuse 0 = fo ll 0.
The numbe e Fom the gphs below, we see tht the slope of the tngent line to the gph of y = 2 t the point ( 0, ) is ppoimtely 0.7 nd y = 3 is ppoimtely.. We should epect tht thee is numbe between 2 nd 3 tht the slope of the tngent line to the gph of y = t ( ) 0, is equl to. In fct, thee is such numbe, nd is denoted by e. Question : Is e polynomil? Invese Functions A function f is sid to be one-to-one if f( ) f( 2) wheneve 2 Tht is, If 2, then f( ) f( 2). Fo emple : - not -
Let f : D R be one-to-one function. Then the invese function defined by f ( y) = f( ) = y fo ny y R This sys tht if f mps to y, then its invese function f : R D is f mps y bck to. Hoizontl Line test A function is one to one if nd only if no hoizontl line intesects its gph moe thn once. Questions :. Wht kind of function hs the invese? 2. If it hs, how to find the invese function? Fo one-to-one function f, how to find the invese function f? Step. Wite y = f( ). Step2. Solve this eqution fo in tems of y. Step3. To epess f s function of, intechnge nd y, the esulting eqution is y = f ( ). Emple : Find the invese of the function Solution : f 3 ( ) 2 = +. 3 Solving the eqution y = + 2 fo in tems of y, we get ( y 2) 3 = Thus, we hve the invese function is ( ) 3 g ( ) = 2. Emple : Wht does it men to invet the sine function? Fist, the sine function clely fils the hoizontl line test.
In ode to hve n invetible function tht tkes ll the vlues of the sine function, we estict the domin of the sine to the intevl fom π to 2 π. 2 This esticted sine nd its invese (clled csine ) e shown in the net figue. f ( ) = sin ( ed) ( ) = sin ( ) f blue Hee e poblems to think bout: π 3π If we estict the domin of sin to, 2 2, wht does the esulting invese function look like? Wht is the eltion between this invese function nd sin? The logithmic function The eponentil function is eithe incesing o decesing nd so it is one-to-one. It hs n invese function which is clled the logithmic function with bse, denoted by log. Hence y log = y =.
Lws of Logithms : log y = log + log y.. ( ) 2. log = log log y 3. log( ) y. = log, whee is ny el numbe. Note tht The logithm with bse e is clled the ntul logithm nd hs specil nottion log = ln. e Hence, we hve ln ( e ) = fo ll. In pticul, ln ( e ) =. Hee e gphs of y = e (ed) nd y ln ( ) = (in blue). Note tht they e mio imges of ech othe though the line y = (in geen).