Chapter 1 Functions PDF

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