MATHEMATICAL METHODS CAS Teach Yourself Series Topic 3: Functions and Relations Inverse Functions, Hybrid Functions, Modulus Functions, Composite Functions and Functional Equations A: Level 4, 474 Flinders Street Melbourne VIC 3000 T: 300 34 58 W: tssm.com.au E: info@tssm.com.au TSSM 20 Page of 8
Contents Functions and Relations... 3 Implied Domain... 3 Absolute value... 3 As it appears in Unit... 3 Inverse functions... 4 As it appears in Unit... 4 Finding rules of inverse functions... 4 Calculator Skills... 4 Hybrid Functions... 5 As it appears in Unit... 5 Reciprocal Functions... 5 As it appears in Unit 3... 5 Modulus graphs... As it appears in Unit 3... Calculator Skills... Addition, Subtraction and Multiplication of ordinates... Calculator skills... Composite functions... Calculator skills... Functional Equations... Calculator skills... Basic Matri Knowledge... Addition and Subtraction... Multiplying a matri by a scalar... Multiplying two matrices together... Determinant of a matri... The inverse of a matri... Calculator Skills... Review Questions... Solutions to Review Questions... 6 TSSM 20 Page 2 of 8
Functions and Relations Implied Domain When a function is stated and the domain is not specified, then the functions is assumed to have its implied domain. This is the largest set of values that belongs to R that the functions can eist over. Eample: If, the implied domain is When answering problems and you are working out the implied domain, remember: You can not have numbers in in the dominator of a fraction that evaluate to 0. You can not have numbers under a square root function that evaluate to a negative number. You can not have numbers that result in a number that is equal to or less than 0 inside a log function. Absolute value As it appears in Unit To undo the absolute value sign when equation solving you use ±. Eample: -4 = 4 4 = 4 Eample: Solve for : + 3 = 5 ( + 3) = 5 3 = 5 = -8 Or + 3 = 5 = 2 TSSM 20 Page 3 of 8
Inverse functions As it appears in Unit Finding rules of inverse functions When you have to find the rule of the inverse function the following process is used:. Rewrite f g into y g. 2. Interchange and y. ie g y 3. Now rearrange the equation so y is the subject (by itself). 4. When you have found the equation rewrite it as f h Note the following:. domf ranf. 2. ranf domf. 3. intercept of f y intercept of f -. 4. y intercept of f intercept of f -. 5. Asymptotes with the equation y k on f = asymptotes with the equation k on 6. Asymptotes with the equation h on f asymptotes with the equation y h on 7. If f and f f f intersect, they intersect on the line y. f. f. Calculator Skills Define functions. Solve f y, y. TSSM 20 Page 4 of 8
Hybrid Functions As it appears in Unit These are functions that have more than one rule that defines them. eg f 2, 0 3, 0 When working out the domain and range we need to be careful. The domain for the above eample is R, however the range is not. The range is, 0 3,. When sketching hybrid functions the following process should be followed:. Draw separate graphs of each separate function. 2. Mark in start and end points that defines each of the separate functions. This is given by the values of that define the function. 3. Draw a separate graph that now shows the rules defined over the appropriate domain. 4. Be sure that you use hollow dots if the end point is not included or filled dots if the end point is included. When looking at domain and range check to see if there are gaps in either or both. Reciprocal Functions As it appears in Unit 3 These are graphs that are written in the form of y =. To sketch these use the basic process of: f(). Sketch y = f() first. 2. Mark in vertical asymptotes. These are where f 0. 3. Now sketch y = f() on all side of asymptotes. You need to take note of the reciprocal behaviour of the function. TSSM 20 Page 5 of 8
. a. = 2 or 2 Solutions to Review Questions Set up 2 equations + 5 = 7 and ( + 5) = 7 b. -2 4 = 24 = -8 or 6 Set up 2 equations 4 = -2 and ( 4) = -2 c. f /5 3 3 set up equation from first step of process = 5y and solve for y. 2. f - () = 3 5 + 5(2 5) Domain: R \ 2 5 Range: R \ 3 5 Use long division to change form of function. set up equation from first step of process and solve for y. TSSM 20 Page 6 of 8
3. Domain: R 0 Range: [3, ) f - () = 2 + 3 y y Intercept ( 0, 3 ) set up equation from first step of process and solve for y 3. Domain: (4, f - () = 3 log e ) Range: R + 4 2 y = -4 0, 3 log e(2) (-2, 0) 3 log e(2), 0) y Intercept ( 0, -2 ) y = -4 Swap and y around as per process and solve for y to get the inverse function TSSM 20 Page 7 of 8
4. a. f(4) = -2 Use first function as 4 is in that domain. b. Range: R ( -2, 0 ) y y Intercept ( 0, 6 ) Intercept ( -2, 0 ) Intercept ( 3, 0 ) Only sketch the section of each rule of the function over the appropriate domain. 5. y y Intercept Local Maimum ( 0, 0.25 ) y = 0 Then reciprocate the y values. Sketch y = 2 + 4 first. TSSM 20 Page 8 of 8