Chapter 1 Functions PDF
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University of Nottingham
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This document is a set of lecture notes from the University of Nottingham, covering functions, a key topic in mathematics and calculus.
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CHAPTER 1 Functions ENGFF033 CALCULUS Learning Outcomes Understand the domain, range, one-one function. Find composite functions. Find the inverse of one-one function. Sketch the graphs of simple functions and its inverse, including piecewise functions. Understand and use...
CHAPTER 1 Functions ENGFF033 CALCULUS Learning Outcomes Understand the domain, range, one-one function. Find composite functions. Find the inverse of one-one function. Sketch the graphs of simple functions and its inverse, including piecewise functions. Understand and use the transformations of functions/graphs 4 Functions Contents 1.1 Definition 1.2 Domain and Range of a Function 1.3 Composite Functions 1.4 Inverse Functions 1.5 Sketching Graphs of Function 1.6 Transformations of Graphs Functions 5 1.1 Definition A function from a set to a set is a rule that assigns a unique (single) element to each Domain: Range: element Set of all Set Any ordered pair is a possible containing relation. input values the range or co-domain Hass, J., Heil, C., & Weir, M. (2019). Thomas' calculus in si units. Pearson Education, Limited. 6 Functions Relation Representing function and its mapping: a) Mapping of P to Q b) A pair of set notation c) Arrow diagram 𝒇 Represented as: **Note: All functions are relations but NOT If and or all relations are e.g. : functions Functions 7 Types o f Rela tion c) One to many relation a) One to one relation Ways to write a function i) ii) iii) A function Not a function b) Many to one relation d) Many to many relation 𝟐 A function Functions Not a function 8 Vertical a nd Horizontal Line Tes t Given as input, as output, Ve r t i c a l L i n e Te s t H o r i z o n t a l L i n e Te s t To determine whether a To determine the relation of a relation is a function. function. Vertical Line intersect the Horizontal Line intersect the graph of at: graph of at a) exactly is a a) exactly is a one- One Point function One Point to-one relation b) > One is NOT b)> One is NOT a Point a function Point one-to-one relation 9 Functions If as input, as output, use vertical line test to determine how many output corresponds to 1 input. State whether it is a function or not. Source: Stewart, J., Clegg, D., & Watson, S. (2023). Calculus (Ninth edition, Metric version.). Cengage Learning. 10 Functions Example 1 Determine if is a function of or not. is a function of is not a function of is not a function of Functions 11 Practice 1 State if the following is the graph of a function. Circle is NOT the Upper semicircle is the Lower semicircle is the graph of a function graph of a function graph of a function 𝟐 𝟐 Functions 12 Practice 2 Practice 3 Given that ( ) Find If ( ) , find when. a) b) c) d) 13 Functions 1.2 Domain & Range of a Function Recall types of function: Linear Where: : variable Function gradient/slope : the constant Quadratic 𝟐 Where: : variable Function : the constants & Root 𝟏 𝒏 or 𝒏 Where: : positive integer greater Function than 1 Reciprocal 𝟏 Where: or Function Rational Where: : numerator Function : denominator & Functions 14 Common rules on Finding i) If involves a Set Solve of possible values of DOMAIN denominator (these values are NOT in the 𝒑(𝒙) domain exclude them). 𝒒(𝒙) ii) If involves an Set Solve inequality for EVEN root must not be 𝟏 𝒏 negative iii) If involves a Set Solve inequality for. logarithm must be 𝒏 positive iv) Others - Domain is thus “all real numbers” Functions 15 Finding i) No specific method for finding the range RANGE of a function. ii) One common method is : Set , solve for and find what values of that is not possible to be excluded from range. iii) Or find the maximum and minimum value of : Range **NOTE: Domain and range of can also be d e t e r m i n e d g r a p h i c a l l y. 16 Functions Finding iv) For rational function, given: RANGE Vertical Horizontal Asymptote Asymptote Condition: degrees of Condition: degrees of & are same < degrees of Range is excluding line that graph never touches horizontal asymptote 17 Functions Finding iii) May use completing the square method for quadratic function: RANGE 𝟐 Express in form of: 𝟐 𝟐 has a minimum value of Range: (Positive) 𝟐 𝒇 has a maximum value of Range: (Negative) 𝟐 𝒇 18 Functions Examples: 𝑦= 1−𝑥 𝑦 = 4−𝑥 19 Functions Exa mple 2 Given for a) Find the domain of the function. b) Sketch the graph of the function. c) Find the range of the function. a) Domain: b) When ( -intercept ) When When c) Range: Pemberton, S., & Gilbey, J. (2018). Cambridge international AS & A level mathematics. Pure mathematics 1. Coursebook. Cambridge University Press. Functions 20 Exa mple 3 For , find its minimum value. Hence, find the range for the domain a) b) 𝟐 𝟐 𝟐 b) Minimum value is still at Minimum Check if min value value at within domain Yes When When When Range: Range: Functions 21 Pra ctice 5 𝟐 𝟐 Find the range of a) for domain b) for domain. Refer Example 3: When Check if min Minimum value at value within domain NO When When When Range: Range: Functions 22 Pra ctice 6 Use completing the square method to find the max/min value and determine the range for each function: a) b) a) b) 𝟐 𝟑 𝟐𝟓 𝟏 𝟐𝟏 Min value at Max value at 𝟐 𝟐 𝟐 𝟒 𝒇 𝒇 Functions 23 Pra ctice 6(c ont.) c) 𝟐 𝟐 c) 𝟐 Min value at 𝒇 Functions 24 Pra ctice 7 ii) When Given 𝟐 for a) Find the domain of the function. b) Sketch the graph of the function. When c) Find the range of the function. a) Domain: 𝟐 b) i) Since minimum value of occurs when Min point c) Range: Pemberton, S., & Gilbey, J. (2018). Cambridge international AS & A level mathematics. Pure mathematics 1. Coursebook. Cambridge University Press. Functions 25 Pra ctice 8 Find the domain and range of the following functions a) b) a) Denominator: b) Domain Domain Range Range 26 Functions Pra ctice 8 (co nt.) c) c ) Domain METHOD 1: Completing the square method for quadratic function: i) Without formula: ii) With formula: 𝟐 𝟐 Min value 𝟏 𝟏𝟑 𝟑 𝟑 Range Min value Range 𝟐 𝟏 𝟏𝟑 𝟏𝟑 𝟏𝟑 𝟑 𝟑 𝟑 𝟑 Functions 27 Pra ctice 8 (co nt.) c) c ) Domain METHOD 2: Differentiation: Min value Range 𝟏𝟑 𝟑 28 Functions Pra ctice 8 (co nt.) d) d) Find domain: Find range: METHOD 2: METHOD 1: METHOD 1: Horizontal Asymptote Denominator Let (degrees of are same) 𝟕 𝟐 Range: METHOD 2: Ve r t i c a l A s y m p t o t e : Domain: 29 Functions 1.3 Comp osite Functions If and are functions, the composite function (" composed with ) is defined by The domain of consists of the numbers in the domain of for which lies in the domain of **NOTE: means ‘first do then do Functions 30 Example 4: Domain of Composite Functions If and f i n d t h e d o m a i n : **NOTE: Find domain a) b) of composite function as well as the domain of a) function inside the composition. b) Domain of composition, : (non-negative) Domain of : Domain of function inside, (non-negative) any values Domain of (non-negative) Functions 31 Example 4 (cont): Domain of Composite Functions If and find the domain: c) d) c) d) 𝟏 𝟒 Domain of : Domain of : (non-negative) any values Domain of Domain of (non−negative) any values Functions 32 Pra ctice 9 Given find the composite functions and the respective domain. a) b) 𝟏 a) 𝒙 𝟐 Domain of : 𝟏 | | | 𝟐 (non-negative) 𝒙 Domain of 33 Functions Pra ctice 9 (co nt.) Given find the composite functions and the respective domain. b) b) 𝟐 | | | Domain of : 𝟐 Domain of 𝟐 (non−negative) 34 Functions Exa mple 5: Finding unknow n func tion in a c omposition Given and , find Solution Functions 35 1.4 Inverse Functions The inverse of a function NOTE: denoted as 𝟏 is 𝟏 𝟏 Identity the function that undoes function what has done. Domain of Range of 𝟏 𝟏 Range of Domain of 𝟏 exists only if is one- to-one function not all function has an inverse. Functions 36 H o r i z o n t a l L i n e Te s t f o r I nve r s e F u n c t i o n Ve r i f i c at i o n If horizontal Line cuts or hits the graph of at Exactly One Point One-to-one relation. 𝟏 Hence it is an inverse function, A above are One-to-one relation, so the inverse 𝟏 exist for each Functions 37 Horizonta l Line Tes t fo r Inver se Func tion Verification A above are Many-to-one relation, so the inverse 𝟏 DOES NOT exist for each Functions 38 Example 6 Show that is the inverse function of Solution is the inverse function of Functions 39 Example 7 Find the inverse function of for METHOD 1 METHOD 2: Identity function STEP 1 Write in terms of STEP 1 Write 𝟏 STEP 2 Swap with STEP 2 Rewrite in terms of STEP 3 Rewrite in terms of 𝟏 𝟐 𝟏 𝟐 40 Functions Pra ctice 10 Given , where domain is. Find the inverse function. STEP 1 Write in terms of Alternative solution 𝟏 STEP 2 Swap with STEP 3 Rewrite in terms of 𝟏 𝟏 41 Functions Example 8: Find domain & range of inverse function Given , where domain is. Find the domain and range of its inverse function. To f i n d R a n g e : From the previous 𝟏 alternative solution: 𝟏 Denominator: Denominator: Range: Domain: 𝟑 𝟑 𝟑 𝟐 𝟐 𝟐 𝟓 𝟓 𝟓 𝟓 𝟓 𝟓 Domain of Range of Range of Domain of 42 Functions Pra ctice 11 Given. Find the inverse function & determine whether it is a function or not. 𝟏 By sketching graph & using vertical test only one intersect point 𝟏 It is a function 43 Functions Pra ctice 12 Given. Find the inverse and the domain of the inverse. METHOD 1: METHOD 2: Denominator 𝟏 𝟏 𝟏 44 Functions Pra ctice 13 Given and Find a) b) c) d) e) a) METHOD 1: METHOD 2: b) 𝟏 𝟏 𝟏 𝟏 45 Functions Pra ctice 13 Given and Find (cont.) a) b) c) d) e) d) e) From (a) &(b) c) 𝟏 𝟏 46 Functions 1.5 Sketching Graphs of Func tion a ) Li n e a r F u nc t i o n 47 Functions Find the range using b ) Q u a d r a t i c F u nc ti o n completing the square method. 𝟐 𝟐 𝟐 𝟐 Max value Min value 48 Functions c ) R oo t F u n c t i o n 𝟒 𝟑 𝟓 Functions 49 d) Reciprocal Function NOTE: Asymptote is a line that curve Ve r t i ca l approaches but a sy mp t o t e does not touch) H o r iz o n t a l a sy m p to t e Functions 50 e) Inverse Function Example: DOMAIN 𝒇 RANGE 𝒇 Find inverse function: 𝟏 𝟏 DOMAIN RANGE of of 𝟏 𝟏 RANGE DOMAIN of 𝟏 of 𝟏 𝟏 **NOTE: 𝟏 𝟏 Since graphs 𝟏 of & are reflections of Functions each other in the line. 51 f ) P i e c e w i s e Fu n c t i o n A function described by using different formulas on different parts of its domain. Example: Closed circle: From the graph of , Open circle: Domain: Range : 52 Functions 1.6 Transform atio ns of Grap hs a ) Tr a n s l a t i on s Ve r t i c a l S h i f t s : If If Horizontal Shifts: If If Functions 53 b ) R e f l e c ti o ns Reflects graph across Reflects graph across Functions 54 c) Scaling Vertical Scaling by a factor of : Stretch Compress Horizontal Scaling by a factor of : Compress Stretch Functions 55 Pra ctice 14 Given , use transformations to graph a) b) c) d) e) Horizontal Shift Vertical Stretch Vertical Shift Reflect Reflect ( ( Source: Stewart, J., Clegg, D., & Watson, S. (2023). Functions Calculus (Ninth edition, Metric version.). Cengage Learning. 56 d ) C o m b i n a ti o ns o f Tr a n s f o r m a t i o n s VERTICAL H O R I Z O N TA L Tr a n s l a t i o n Tr a n s l a t i o n Reflection Reflection across across Ve r t i c a l s t r e t c h Horizontal stretch Functions 57 Exa mple 9 Sketch the graphs below a) Horizontal compress a) b) Source: Stewart, J., Clegg, D., & Watson, S. (2023). Calculus (Ninth edition, Metric version.). Cengage Learning. Functions 58 Exa mple 9 (cont.) Sketch the graphs below a) b) b) From First, reflect across Next, from Vertical shift upwards Source: Stewart, J., Clegg, D., & Watson, S. (2023). Calculus (Ninth edition, Metric version.). Cengage Learning. Functions 59 Pra ctice 15 Using graph transformations method, First, from sketch the graph of the function 𝟐 Horizontal shift left Apply completing the square method: 𝟐 𝟐 𝟐 Next, from 𝟐 𝟐 Vertical shift upwards 𝟐 𝟐 Source: Stewart, J., Clegg, D., & Watson, S. (2023). Functions Calculus (Ninth edition, Metric version.). Cengage Learning. 60 Pra ctice 16 Figure 1 shows function, find the function expression to a) compress the graph horizontally by a factor of 2 followed by a reflection across the as shown in Figure 2. Figure 2 a) Horizontal compress Reflect ( 𝟒 𝟑 Figure 1 Functions 61 Pra ctice 16(cont.) b) compress the graph vertically by a factor of 2 followed by a reflection across the as shown in Figure 3. Figure 3 b) Vertical compress Reflect ( Figure 1 𝟒 𝟑 Functions 62 Recap D o m a i n , ra n g e , o n e - o n e f u n c t i o n. D o m a i n & ran g e f o r q u a d ra t i c f u n c t i o n composite functions i nve r s e f u n c t i o n. G ra p h s s ke t ch i n g Tra n s f o r m at i o n s o f fu n c t i o n s/ g rap h s Functions 63