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  Math 150 Exam 1 Review Sheet Function:  A function is a mapping that assigns each element  x  in a set  D , called the domain,to  exactly one  element in a set  R , called the range. Domain and Range:  Given a function  f  ( x ) , the domain is the set of values which can beplugged in for  x  (common restrictions to the domain include values that result in zerodenominators or negatives under even radicals). The range is the set of all possible answers( y -values). Composition of Functions:  The composition of   f   and  g  is given by  ( f  ◦ g )( x ) =  f  ( g ( x )) .The domain includes all  x  in the domain of   g  such that  g ( x )  is in the domain of   f  . In practice,we find the domain by finding the domain of the inside function and of the final result, andthen taking the intersection of these two. Exponential Functions:  The function  f  ( x ) =  a x with  a >  0  has the following generalshapes: a >  1 a <  1     (0 , 1) One-to-One Function:  A function  f   is 1:1 no two  x -values are assigned to the same  y -value,ie.  f  ( x 1 )  =  f  ( x 2 )  whenever  x 1   =  x 2 . Inverse Function:  If   f   is a 1:1 function with domain  A  and range  B , there is an inversefunction  f  − 1 ( x )  with domain  B  and range  A  defined by any of the following:i)  f  − 1 ( y ) =  x ⇔ f  ( x ) =  y ;ii)  f  − 1 ( f  ( x )) =  x  for every  x  in  A  and  f  ( f  − 1 ( x )) =  x  for every  x  in  B ;iii)  f  − 1 ( x )  is the reflection of   f  ( x )  through the line  y  =  x . Steps to find  f  − 1 :  (1) Rewrite  f  ( x )  as  y ; (2) Switch all  x ’s and  y ’s; (3) Solve for  y ;(4) Rewrite  y  as  f  − 1 ( x )  Logarithmic Functions:  The  log  function is defined as the inverse of the exponentialfunction, and therefore satisfies  log a  x  =  y  ⇔ a y =  x  and has domain  (0 , ∞ )  and range ( −∞ , ∞ ) . Since  log  and exponentials are inverses, we have  log a ( a x ) =  x  and  a log a  x =  x . If the base of the logarithm is  e , we write  ln x , called the natural logarithm. Limit as  x  approaches  a :  We write  lim x → a  f  ( x ) =  L  if as  x  gets closer to  a  from either sideof   a  (but unequal to  a ),  f  ( x )  becomes arbitrarily close to  L . If   f  ( x )  becomes arbitrarily largeor small as  x  approaches  a , we write  lim x → a  f  ( x ) = ∞ and  lim x → a  f  ( x ) = −∞ , respectively.For the limit to exist, the limits from the left ( x < a ) and the right ( x > a ) must be equal, ie. lim x → a −  f  ( x ) = lim x → a +  f  ( x ) . Otherwise, the limit does not exist. Evaluating  lim x → a  f  ( x ) :  For polynomials and rational functions in which  a  is in the domain,we can evaluate the limit by simply replacing  x  by  a , ie.  lim x → a  f  ( x ) =  f  ( a ) . For rationalfunctions that have vertical asymptotes at  a , we must consider the signs as the denominatorapproaches 0.* Definition of Limit:  We say  lim x → a  f  ( x ) =  L  if for every  ǫ >  0 , there is  δ >  0  such that 0  < | x − a | < δ  ⇒| f  ( x ) − L | < ǫ . Continuity:  A function  f   is continuous at  a  if   lim x → a  f  ( x ) =  f  ( a ) . The three types of discontinuities are holes (removable discontinuities), gaps (jump discontinuities), and verticalasymptotes (infinite discontinuities).* The Intermediate Value Theorem:  Suppose that  f   is continuous on the interval  [ a,b ]  and f  ( a )  =  f  ( b ) . Then for every number  N   between  f  ( a )  and  f  ( b ) , there is a number  c  satisfying a < c < b  and  f  ( c ) =  N  . Limits at Infinity:  We say  lim x →∞ f  ( x ) =  L  if as  x  grows larger,  f  ( x )  becomes arbitrarilyclose to  L . The line  f  ( x ) =  L  is then a horizontal asymptote. The definition for  x approaching −∞ is similar. Shortcuts for Computing Limits at Infinity:  Given a rational function  f  ( x ) =  p ( x ) /q  ( x ) ,we have the following rules:i)  lim x →±∞ f  ( x ) = 0  if   degree (  p )  < degree ( q  ) ;ii)  lim x →±∞ f  ( x ) =  Ratio of Leading Coefficients  if   degree (  p ) =  degree ( q  ) ;iii)  lim x →±∞ f  ( x ) = ±∞ if   degree (  p )  > degree ( q  ) , where the resulting sign must bedetermined based on the behavior of the leading terms of   p  and  q  .  Asymptotes:  For a rational function, vertical asymptotes (of the form  x  =  k ) occur at thezeros of the denominator, provided there is no cancellation of the corresponding factor. Forexample,  f  ( x ) =  x − 2 x 2 − 4  =  x − 2( x − 2)( x +2)  =  1 x +2  has a vertical asymptote only at  x  = − 2 .Horizontal asymptotes are given by  y  = lim x →±∞ f  ( x ) .* Slopes and Derivatives:  The slope of the tangent line to the function  f  ( x )  at  x  =  a  is givenby the derivative  f  ′ ( a ) . This is formally computed as f  ′ ( a ) = lim h → 0 f  ( a + h ) − f  ( a ) h  .* The Derivative as a Function:  The derivative of the function  f  ( x )  is formally computed as f  ′ ( x ) = lim h → 0 f  ( x + h ) − f  ( x ) h  . Average and Instantaneous Velocity  If position is given by the function  s ( t ) , theinstantaneous velocity at any time  t  is given by the function  v ( t ) =  s ′ ( t ) .*Formula will be provided.
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