Higher. Functions and Graphs. Functions and Graphs 15

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1 Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 5 Set Theor 5 Functions 6 Inverse Functions 9 4 Eponential Functions 0 5 Introduction to Logarithms 0 6 Radians 7 Eact Values 8 Trigonometric Functions 9 Graph Transformations HSN00 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For more details about the copright on these notes, please see

2 Higher Mathematics Unit Mathematics UTCME Functions and Graphs Set Theor In order to stud functions and graphs, we use set theor. This requires some standard smbols and terms, which ou should become familiar with. set element subset a collection of objects (usuall numbers (or member an object which is part of a set a set which is part of another set { 5, 6, 7, 8 } is a set belongs to; is a member of 6 { 5, 6, 7, 8} does not belong to; is not a member of 7 is an element of { 5, 6, 7, 8 } { 5, 6 } is a subset of { 5, 6, 7, 8 } 4 { 5, 6, 7, 8} { } or the empt set a special set with no members Standard Sets There are common sets of numbers which have their own smbols. Note that numbers can belong to more than one set. N natural numbers counting numbers, i.e. N = {,,, 4, 5,...} W whole numbers natural numbers including zero, i.e. W = { 0,,,, 4,...} Z integers positive and negative whole numbers, i.e. Z = {,,, 0,,, } Q rational numbers can be written as a fraction of integers, e.g. 4,, 0 5, R real numbers all points on the number line, e.g. 6,,,, 0 5 Page 5 HSN00

3 Higher Mathematics Unit Mathematics Notice that N is a subset of W, which is a subset of Z, which is a subset of Q, which is a subset of R. These relationships between the standard sets are illustrated in the Venn diagram below. R Q Z W N EXAMPLE List all the numbers in the set P = { N :< < 5}. P contains natural numbers which are strictl greater than and strictl less than 5, so: P = {,, 4} Note In Set Theor, a colon ( : means such that Functions A function is a rule which connects a set of numbers to another set. The set of starting numbers is called the domain and the resulting set is called the range. f f ( domain range A function is usuall denoted b a lower case letter (e.g. f or g and is defined using a formula of the form f ( =. Restrictions on the Domain The domain of a function must be defined such that the function can be evaluated for all elements of the domain. In other words, the domain can onl contain numbers which give answers when worked through the function. Division b Zero It is impossible to divide b zero, so in functions involving fractions, the domain must eclude numbers which would give a denominator (bottom line of zero. Page 6 HSN00

4 Higher Mathematics Unit Mathematics For eample, the function defined b: f ( = 5 cannot have 5 in its domain, since this would make the denominator equal to zero. The domain of f ma be epressed formall as { R : 5}. This is read as all belonging to the real set such that does not equal five. Even Roots Using real numbers, we cannot evaluate an even root (i.e. square root, fourth root etc of a negative number. So an functions involving even roots must eclude numbers which would give a negative number under the root. For eample, the function defined b: f ( = 7 must have 7 0. Solving for gives 7, so the domain of f can be epressed formall as { : } EXAMPLE R A function g is defined b g ( =. + 4 Define a suitable domain for g. We cannot divide b zero, so 4 Identifing the Range R.. So the domain is { : 4} Some functions cannot produce certain values so these are not in the range. For eample: f ( = does not produce negative values, since an number squared is either positive or zero. Looking at the graphs of functions makes identifing the range more straightforward. = f ( If we consider the graph of = f ( (shown to the left it is clear that there are no negative -values. The range can be stated formall as f ( 0. Page 7 HSN00

5 Higher Mathematics Unit Mathematics EXAMPLE. A function f is defined b f ( = sin for R. Identif its range. Sketching the graph of = f ( shows that sin onl produces values from to inclusive. = sin This can be written as f (. Composite Functions Functions can be combined to give a composite function. If we have two functions defined b f ( and g (, then f g ( and g f ( define composite functions. In most cases f g ( g f. f f ( g h g( f or h EXAMPLES. Defined on suitable domains, f ( = and g ( =. Find: (a f (b f g ( (c g f ( (a f = = 4 = ( (b f g ( f = ( = ( (c g f ( g = 4. Functions f ( = + and g ( = are defined on suitable domains. Find formulae for h ( = f g ( and k ( = g f (. h ( = f g ( = f ( = + k ( = g f ( = g + = + Page 8 HSN00

6 Higher Mathematics Unit Mathematics Inverse Functions Defined on suitable domains, all the functions we will meet have an inverse function, which reverses the effect of the function. If we have a function defined b f (, its inverse is usuall denoted f (. If a number is worked through a function f then a function g, and the result is the same as ou started with, i.e. g f ( =, then f and g are inverses. + For eample, f ( = 4 and g ( = are inverse functions. If we 4 work the number through f and then the result through g, we will get back again. + f = 4 g = 4 = = Graphs of Inverses If we have the graph of a function, then we can find the graph of its inverse b reflecting in the line =. For eample, the diagrams below show the graphs of two functions and their inverses. = f ( = f f ( g = g ( = = f ( = g ( Page 9 HSN00

7 Higher Mathematics Unit Mathematics 4 Eponential Functions An eponential function is one in the form f ( = a where a, R and a > 0. This is known as an eponential function to the base a; is referred to as the power, inde or eponent. 0 Notice that when = 0, f ( = a =. Also when =, f ( = a = a. Hence the graph of an eponential alwas passes through ( 0, and (, a : = a, a > = a, 0 < a < (, a (, a EXAMPLE Sketch the curve with equation = 6. The curve passes through ( 0, and (, 6. = 6 (, 6 5 Introduction to Logarithms Until now, we have onl been able to solve problems involving eponentials when we know the inde, and have to find the base. For eample, we can 6 solve k = 5 b taking sith roots to get k = 6 5. But what if we know the base and have to find the inde? To solve 6 k = 5 for k, we need to find the power of 6 which gives 5. To save writing this each time, we use the notation k = log6 5, read as log to the base 6 of 5. In general: log a is the power of a which gives The properties of logarithms will be covered in Unit utcome. Page 0 HSN00

8 Higher Mathematics Unit Mathematics Logarithmic Functions A logarithmic function is one in the form f ( = log a where a, > 0. Logarithmic functions are inverses of eponentials, so to find the graph of = log a, we can reflect the graph of = a in the line =. = log a ( a, The graph of a logarithmic function alwas passes through (, 0 and ( a,. EXAMPLE Sketch the curve with equation = log6. The curve passes through (, 0 and ( 6,. = log 6 ( 6, 6 Radians Degrees are not the onl units used to measure angles. The radian (RAD on the calculator is a measurement also used. Degrees and radians are related such that: π radians = 80 The other equivalences that ou should be familiar with are: 0 = π 6 radians 45 = π radians 60 = π 4 radians 90 = π radians 5 = π 4 radians 60 = π radians Converting between degrees and radians is straightforward. To convert from radians to degrees, multipl b 80 and divide b π. Degrees To convert from degrees to radians, multipl b π and divide b 80. For eample, 50 = 50 π 5 80 = 8π radians. 80 π 80 π Radians Page HSN00

9 Higher Mathematics Unit Mathematics 7 Eact Values The following eact values must be known. You can do this b either memorising the two triangles involved, or memorising the table. DEG RAD sin cos tan π 6 45 π 4 60 π 90 π 0 8 Trigonometric Functions Periodic functions have a repeating pattern in their graphs. The length of the smallest repeating pattern in the -direction is called the minimum period. If the repeating pattern has a minimum and maimum value, then half of the difference between the minimum and maimum is called the amplitude. ma. value amplitude min. value minimum period The three basic trigonometric functions (sine, cosine, and tangent are periodic, and have graphs as shown below. = sin = cos = tan Period = 60 = π radians Amplitude = Period = 60 = π radians Amplitude = Period = 80 =π radians Amplitude is undefined Page HSN00

10 Higher Mathematics Unit Mathematics 9 Graph Transformations The graphs below represent two functions. ne is a cubic and the other is a sine wave, focusing on the region between 0 and 60. There are three different things we can do to the graphs. Translation ( p, q = g A translation moves ever point on a graph a fied distance in the same direction. The shape of the graph does not change = sin Translation parallel to the -ais f ( + a moves the graph of f ( up or down. The graph is moved up if a is positive, and down if a is negative. (, a is positive ( p, q + = g ( + (, = sin + (, a is negative ( p, q = g (, = sin Page HSN00

11 Higher Mathematics Unit Mathematics Translation parallel to the -ais f ( + a moves the graph of f ( left or right. The graph is moved left if a is positive, and right if a is negative. a is positive a is negative = g ( + = g ( ( p, q ( p +, q 5 = sin( + 90 = sin( Reflection When reflecting, the graph is flipped about one of the aes. It is important to appl this transformation before an translation. Reflection in the -ais f ( reflects the graph of f ( in the -ais. = g ( ( p, q = sin Reflection in the -ais f ( reflects the graph of f ( in the -ais. ( p, q = g ( = sin( From the graphs, sin( = sin Page 4 HSN00

12 Higher Mathematics Unit Mathematics Scaling Scaling verticall kf ( scales the graph of f ( in the vertical direction. The -coordinate of each point on the graph is multiplied b k, roots are unaffected. These eamples consider positive k. k > 0 < k < ( p, q = g ( ( p, q = g ( = sin Negative k causes the same scaling, but the graph must then be reflected in the -ais: = g ( = sin ( p, q Page 5 HSN00

13 Higher Mathematics Unit Mathematics Scaling horizontall f ( k scales the graph of f ( in the horizontal direction. The coordinates of the -ais intercept sta the same. The eamples below consider positive k. k > 0 < k < = g ( = g ( p, q ( p, q 6 = sin = sin Negative k causes the same scaling, but the graph must then be reflected in the -ais: ( p, q = g ( Page 6 HSN00

14 Higher Mathematics Unit Mathematics EXAMPLES. The graph of = f ( is shown below. = f ( 0 (, 4 Sketch the graph of = f (. 5 Reflect in the -ais, then shift down b : = f ( ( 5, 0 (, 6. Sketch the graph of = 5cos( where = 5cos( Page 7 HSN00