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Higher Maths 1 2 Functions1

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The symbol means is an element of. Introduction to Set Theory In Mathematics, the word set refers to a group of numbers or other types of elements. Sets are written as follows: Examples { 1, 2, 3, 4, 5, 6 }{ -0.7, -0.2, 0.1 }{ red, green, blue } 4 { 1, 2, 3, 4, 5 } 7 { 1, 2, 3 } { 6, 7, 8 } { 6, 7, 8, 9 } If A = { 0, 2, 4, 6, 8, … 20 } and B = { 1, 2, 3, 4, 5 } then B A Sets can also be named using letters: P = { 2, 3, 5, 7, 11, 13, 17, 19, 23, … } Higher Maths 1 2 1 Sets and Functions2

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N W Z { 1, 2, 3, 4, 5,... } { 0, 1, 2, 3, 4, 5,... } {... -3, -2, -1, 0, 1, 2, 3,... } The Basic Number Sets Q Rational numbers Includes all integers, plus any number which can be written as a fraction. R 7 π Includes all rational numbers, plus irrational numbers such as or. Real numbers C Complex numbers Includes all numbers, even imaginary ones which do not exist. Whole numbers Integers Natural numbers N W ZQ RC 2 3 Q -1 R Examples Higher Maths 1 2 1 Sets and Functions3

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Set Theory and Venn Diagrams Venn Diagrams are illustrations which use overlapping circles to display logical connections between sets. Blue Animal Food Pig Blueberry Pie Blue Whale ? Rain Red Yellow Orange Juice Sun Strawberries Aardvark N W Z Q R C 82 5 7 Higher Maths 1 2 1 Sets and Functions4

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Function Domain and Range Any function can be thought of as having an input and an output. The input is sometimes also known as the domain of the function, with the output referred to as the range. f (x)f (x) domain range Each number in the domain has a unique output number in the range. The function has the domain { -2, -1, 0, 1, 2, 3 } Find the range. Imporant Example f ( x ) = x 2 + 3 x f ( - 2 ) = 4 – 6 = - 2 f ( - 1 ) = 1 – 3 = - 2 f ( 0 ) = 0 + 0 = 0 f ( 1 ) = 1 + 3 = 4 f ( 2 ) = 4 + 6 = 10 Range = { -2, 0, 4, 10 } Higher Maths 1 2 1 Sets and Functions5

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Composite Functions It is possible to combine functions by substituting one function into another. f (x)f (x) g (x)g (x) g ( ) f (x)f (x) is a composite function and is read. g ( ) f (x)f (x) g of f of x Important g ( ) f (x)f (x) f ( ) g (x)g (x) In general Given the functions Example g ( x ) = x + 3 f (x) = 2 xf (x) = 2 x and find and. = 2 ( ) x + 3 = 2 x + 6 = ( ) + 3 2 x2 x = 2 x + 3 f ( g ( x )) g ( f ( x )) f ( g ( x )) Higher Maths 1 2 1 Sets and Functions6

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= x= x Inverse of a Function If a function also works backwards for each output number, it is possible to write the inverse of the function. f (x)f (x) f (x) = x 2f (x) = x 2 f ( 4 ) = 16 f ( -4 ) = 16 Not all functions have an inverse, e.g. Every output in the range must have only one input in the domain. does not have an inverse function. f (x) = x 2f (x) = x 2 f ( 16 ) = ? - 1 domain range Note that f ( ) - 1 f (x)f (x) f ( x ) - 1 = x= x x and f ( ) f ( x ) - 1 Higher Maths 1 2 1 Sets and Functions7

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Find the inverse function for. Finding Inverse Functions g ( x ) = 5 x 3 – 2 - 1 Example g (x)g (x) 3 × 5× 5 – 2– 2 g x x + 2 3 ÷ 5÷ 5 + 2+ 2 g - 1 x x x + 2 5 5 3 - 1 g ( x ) = + 2+ 2 ÷ 5÷ 5 3 Higher Maths 1 2 1 Sets and Functions8

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Basic Functions and Graphs f ( x ) = ax f ( x ) = a sin bx f ( x ) = a tan bx f ( x ) = ax ² f ( x ) = ax ³ f ( x ) = a x Linear Functions Quadratic Functions Trigonometric Functions Cubic FunctionsInverse Functions Higher Maths 1 2 1 Sets and Functions9

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Finding Equations of Exponential Functions Higher Maths 1 2 1 Sets and Functions10 It is possible to find the equation of any exponential function by substituting values of and for any point on the line. y = a + b x 2 (3,9)(3,9) Example The diagram shows the graph of y = a + b Find the values of a and b. Substitute (0,2): x 2 = a + b 0 = 1 + b b = 1 xy Substitute (3,9): 9 = a + 1 3 a = 2 a = 8 3 y = 2 + 1 x

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Graphs of Inverse Functions To sketch the graph of an inverse function, reflect the graph of the function across the line. f ( x ) - 1 f (x)f (x) y = xy = x y = xy = x - 1 f (x)f (x) y = xy = x g ( x ) - 1 g (x)g (x) Higher Maths 1 2 1 Sets and Functions11

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Exponential and Logartithmic Functions f ( x ) = log x a 1 f ( x ) = a x 1 (1,a)(1,a) (a,1 ) is called an exponential function with base. Exponential Functions f ( x ) = a x a The inverse function of an exponential function is called a logarithmic function and is written as. f ( x ) = log x a Logarithmic Functions Higher Maths 1 2 1 Sets and Functions12

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