RI Functions Solns

RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 10 (Summary and Tutorial) Topic: Functions Summary for Functions Relations and Functions  The set of inputs of a function is known as the domain while the corresponding set of outputs is known as the range.  To define a function completely, both the rule and domain must be specified.  The range of a function can be obtaining by looking at its graph (y-values).  If a function is such that no two elements in the given domain have the same output, it is a one- one function.  To check if a function is one-one, we use horizontal lines to check. If function is one-one, such as 2f: , 2xx x  , we write since every horizontal line y = k,  4,k , cuts the graph of f at one and only one point, f is a one-one function. Note that  4, is the range of f. If function is NOT one-one, such as 2f: ,xx x  , we choose a specific line y = k where k is a value in range of f. In this case, we can write since the horizontal line y = 1 cuts the graph of f at more than one point, f is not a one-one function Inverse Functions  Given a function f, the inverse function 1f exists if and only if f is one-one.  To define 1f , you must find the rule and domain.  To find the rule of 1f , let f( )yx and make x the subject, i.e -1f()xy . Example to find the rule of inverse of g: 1 e , xxx  Let 1e xy  and make x the subject to get ln( 1)xy  .  To find domain of 1f , use the result: 1fD   fR Note also that 1 ffRD  .  The graphs of f( )yx and 1f( )yx  are reflections of each other in the line yx .

Composite Functions  Given 2 functions f and g, the composite function gf exists if and only if fRD g .  To define gf, you must find the rule and domain.  To find the rule of gf, find f( )x first, then find gf () x .  To find domain of gf, use the result: gfD  fD  In general, fg gf . i.e. composition of functions is not commutative.  The composite function ff, if it exists, is written as 2f , i.e.  2f ( )f f ( )f f ( )xx x . o Useful results: (i) 1ff ( )xx  , where 1fDx  . (ii) 1ff ( ) xx  , where fDx .  To find the range of gf, use either ( 1 ) gfR  gR restricted to fR or (2) sketch the graph of gf ( )yx over gfD and find gfR . Example Find the range of the composite gf given that the functions f and g are defined as follows: f : 2, 3xx x  , g : , 0xxx  . Method 1 Sketch the graphs of the functions f and g on separate diagrams. For the composite function gf, f is first applied to each fDx to find the image f( )x , and then the rule for g is applied to f( )x to obtain the image gf ()x . i.e. g f () gf ()xx . Consequently, we can obtain the range of gf as the range of g whose domain is restricted range of f, i.e gf[