Chapter XX: 1: Functions. XXXXXXXXXXXXXXX <CT>Chapter 1: Data representation</ct> 1.1 Mappings

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1 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter XX: : Functions XXXXXXXXXXXXXXX <CT>Chapter : Data representation</ct> This section will show you how to: understand and use the terms: function, domain, range (image set), one-one function, inverse function and composition of functions use the notation f() = +, f :, f () and f () understand the relationship between y = f() and y = f() solve graphically or algebraically equations of the type a + b = c and a + b = c + d eplain in words why a given function is a function or why it does not have an inverse find the inverse of a one-one function and form composite functions use sketch graphs to show the relationship between a function and its inverse.. Mappings The table below shows one-one, many-one and one-many mappings. one-one many-one one-many y y f() = ± f() = + O f() = O O For one input value there is just one output value. For two input values there is one output value. For two input value there are two output values. Eercise. Determine whether each of these mappings is one-one, many-one or one-many. + R + R R R R, > R, 0 7 R, > 0 8 ± R, 0

2 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Cambridge IGCSE and O Level Additional Mathematics. Definition of a function A function is a rule that maps each value to just one y value for a defined set of input values. This means that mappings that are either one-one many-one are called functions. The mapping + where R, is a one-one function. The function can be defined as f : +, R or f() = +, R. The set of input values for a function is called the domain of the function. The set of output values for a function is called the range (or image set) of the function. worked eample The function f is defined by f()= ( ) + for 0. Answers f()= ( ) + is a positive quadratic function so the graph will be of the form ( ) + The minimum value of the epression is 0 + = and this minimum occurs when =. So the function f()= ( ) + will have a minimum point at the point (, ). When = 0, y = ( 0 ) + =. This part of the epression is a square so it will always be 0. The smallest value it can be is 0. This occurs when =. When =, y = ( ) + = 0. y (, 0) Range The range is f() 0. O (, ) Domain Eercise. Which of the mappings in Eercise. are functions? Find the range for each of these functions. a f() = 9, 8 b f() =, 0 6 c f() = 7, d f() =, e f() =, f f ( ) =, 6 The function g is defined as g( ) = for 0. Find the range of g. The function f is defined by f( ) = for R.

3 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter : Functions The function f is defined by f( ) = ( ) for. 6 The function f is defined by f( ) = ( + ) for. 7 The function f is defined by f: 8 ( ) for 7. 8 The function f is defined by f( ) = for. 9 Find the largest possible domain for the following functions. a f ( ) = b f ( ) = c + ( )( + ) d f ( ) = e f: f f: + g g: h f: i f:. Composite functions When one function is followed by another function, the resulting function is called a composite function. fg() means the function g acts on first, then f acts on the result. f () means ff(), so you apply the function f twice. worked eample f: + for R g: for R Find fg(). Answer fg() g acts on first and g() = =. = f() = + = worked eample g()= for R h() = for R Solve the equation hg( ) =.

4 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Cambridge IGCSE and O Level Additional Mathematics Answers hg( ) g acts on first and g() =. ( ) ( ) = h h is the function triple and take from. = Epand the brackets. = = 0 6 hg( ) = = 0 6 = 6 = =± Set up and solve the equation. Eercise. f( ) = for R g( ) = + for R Find the value of gf(). f( ) = ( ) for R Find f ( ). The function f is defined by f( ) = + for. The function g is defined by g( ) = for > 0. Find gf(7). The function f is defined by f( ) = ( ) + for >. The function g is defined by g + ( ) = for >. + Find fg(6). f: for > 0 g: for > 0 Epress each of the following in terms of f and g. a b 6 The function f is defined by f: for R. 8 The function g is defined by g: for. Solve the equation gf() =. 7 f( ) = + for > 0 g( ) = for > 0 Solve the equation fg() =. 8 The function f is defined, for R, by f:,. The function g is defined, for R, by + g:. Solve the equation fg() =. TIP Before writing your final answers, compare your solutions with the domains of the original functions.

5 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter : Functions 9 The function g is defined by g( ) = for 0. The function h is defined by h( ) = for 0. Solve the equation gh( ) = giving your answer(s) as eact value(s). 0 The function f is defined by f: for R. The function g is defined by g: + for R. Epress each of the following as a composite function, using only f and g. a ( + ) b + c + d The functions f and g are defined for > 0 by f: + and g: Epress in terms of f and g a + b + 6 c + Given the functions f ( ) = and g( ) = a Find the domain and range of g. b Solve the equation g( ) = 0. c Find the domain and range of fg.. Modulus functions +, The modulus (or absolute value) of a number is the magnitude of the number without a sign attached. The modulus of, written as, is defined as if > 0 = 0 if = 0 if < 0 The statement, = k, where 0, means that = k or = k. worked eample a + = + 8 b 9 = 7 Answers a + = = + 8 = = Solution is : = or b 7 = 9 7 = 9 = 6 = 8 =± Solution is : =± or + = 8 = = or 7 = 9 = =

6 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Cambridge IGCSE and O Level Additional Mathematics Eercise. Solve. a = b + = 8 c 6 = d g = 6 e + = f 9 = + 6 = h + = i 6 = TIP Remember to check your answers to make sure that they satisfy the original equation. Solve. a + = b = c + + = d = e + = f 7 = Solve giving your answers as eact values if appropriate. a = b + = c 9 = d = e 6 = + f = + g = h = i + = 6 Solve each of the following pairs of simultaneous equations. a y = + b y = y = y =. Graphs of y = f() where f() is linear Eercise. Sketch the graphs of each of the following functions showing the coordinates of the points where the graph meets the aes. a y = b y = c y = d y = e y = 6 f y = a Complete the table of values for y =. 0 y b Draw the graph of y = for. Draw the graphs of each of the following functions. a y = + b y = c y = d y = + e y = 6 f y = Given that each of these functions is defined for the domain, find the range of a f: 6 b g: 6 c h: 6.

7 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter : Functions a f: for b g: for c h: for Find the range of each function for. 6 a Sketch the graph of y = for < <, showing the coordinates of the points where the graph meets the aes. b On the same diagram, sketch the graph of y = +. c Solve the equation = +. 7 A function f is defined by f( ) =, for. a Sketch the graph of y = f(). b State the range of f. c Solve the equation f( ) =. 8 a Sketch on a single diagram, the graphs of + y = 6 and y = +. b Solve the inequality + < ( ) 6..6 Inverse functions The inverse of the function f() is written as f (). The domain of f () is the range of f(). The range of f () is the domain of f(). It is important to remember that not every function has an inverse. An inverse function f () can eist if, and only if, the function f() is a one-one mapping. 7 worked eample f() = ( + ) for > a Find an epression for f (). b Solve the equation f () =. Answers a f() = ( + ) for > Step : Write the function as y = y = ( + ) Step : Interchange the and y variables. ( y ) = + Step : Rearrange to make y the subject. + = ( y + ) + = y + y = + f ( ) = + b f ( ) =. + = + = 6 + = 6 =

8 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Cambridge IGCSE and O Level Additional Mathematics Eercise.6 f( ) = ( + ) for. Find an epression for f (). f ( ) = for 0. Find an epression for f (). f( ) = ( ) + for. Find an epression for f (). f( ) = for. Find an epression for f (). f: for > 0 g: Epress f () and g () in terms of. for. 8 6 f( ) = ( ) + for > a Find an epression for f (). b Solve the equation f ( ) = f(. ) 7 g + ( ) = for > a Find an epressions for g () and comment on your result. b Solve the equation g () = 6. 8 f ( ) = for R g( ) = for R a Find f (). b Solve fg( ) = f ( ) leaving answers as eact values. 9 f: + for g: Solve the equation f( ) = g ( ). for > 0 If f ( ) = 9 + R find an epression for f (). If f ( ) = and g( ) =, solve the equation f g( ) = 0.0. Find the value of the constant k such that f ( ) = is a self-inverse function. + k The function f is defined by f ( ) =. Find an epression for g() in terms of for each of the following: a fg( ) = + b gf ( ) = + TIP A self-inverse function is one for which f() = f (), for all values of in the domain.

9 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter : Functions + Given f( ) = + and g ( ) = find the following. a f b g c ( fg) d ( gf ) e f g f g f Write down any observations from your results. + Given that fg ( ) = and g( ) = + find f(). 6 Functions f and g are defined for all real numbers. g( ) = + 7 and gf ( ) = Find f()..7 The graph of a function and its inverse The graphs of f and f are reflections of each other in the line y =. This is true for all one-one functions and their inverse functions. This is because: ff ( ) = = f f ( ). y 6 y = f f 9 O 6 Some functions are called self-inverse functions because f and its inverse f are the same. If f ( ) = for 0, then f ( ) = for 0. So f ( ) = for 0 is an eample of a self-inverse function. When a function f is self-inverse, the graph of f will be symmetrical about the line y =. Eercise.7 On a copy of the grid, draw the graph of the inverse of the function = y. y = y 6 O 6 8

10 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Cambridge IGCSE and O Level Additional Mathematics f( ) = +, 0. On the same aes, sketch the graphs of y = f() and y = f (), showing the coordinates of any points where the curves meet the coordinate aes. g( ) = for 0. Sketch, on a single diagram, the graphs of y = g( ) and y = g ( ), showing the coordinates of any points where the curves meet the coordinate aes. The function f is defined by f( ) = 6 for all real values of a Find the inverse function f (). b Sketch the graphs of f() and f () on the same aes. c Write down the point of intersection of the graphs f() and f (). Given the function f( ) = for. a Eplain why f () eists and find f (). b State the range of the function f (). c Sketch the graphs of f() and f () on the same aes. d Write down where f () crosses the y ais. 6 a By finding f () show that f ( ) = function. R, is a self-inverse 0 b Sketch the graphs of f() and f () on the same aes. c Write down the coordinates of the intersection of the graphs with the coordinate aes. Summary Functions A function is a rule that maps each -value to just one y-value for a defined set of input values. Mappings that are either one-one many-one are called functions. The set of input values for a function is called the domain of the function. The set of output values for a function is called the range (or image set) of the function. Modulus function The modulus of, written as, is defined as if > 0 = 0 if = 0 if < 0 Composite functions fg() means the function g acts on first, then f acts on the result. f () means ff().