1.1 Functions Inverse Functions Questions - The …

1 Gr 12 Maths Functions GR 12 MATHS. Functions . Questions and ANSWERS. Work through the Grade 11 Functions downloads first to ensure your foundation is solid before attempting Inverse Functions . We wish you the best of luck for your exams. From The Answer Series team Copyright The Answer 1. Gr 12 Maths Functions : Questions 2. Consider the function f (x) = - 3x + 6. Is g the Inverse function of f ? Why (not)? (2). Write down the domain and range of f. (a) Name 2 ways in which the domain of f could be Inverse Functions Determine the equation of the Inverse of f in the form (2) restricted to ensure that the Inverse is a function. Sketch the 2 cases. (4). (Gr 12 only) f -1 (x) = .. (2). (b) Determine f -1 (x). in each case. Sketch the 2 cases. Sketch the graphs of the Functions f, f -1 and y = x on Include the line y = x. (4). Questions the same set of axes. What do you notice? (3). (c) Determine the domain and the range of f -1 in each -1 (3).

2 If f (1) = 3, then f = ..? (1). case. (4). f (2) = 0 ( ; ) lies on f and ( ; ) lies on f -1 . (2). 2. 7. Consider the function f where f (x) = - x and x m 0. Concepts and techniques involving the general Write down the coordinates of the x- and y-intercepts of the Write down the domain and range of f. (2). characteristics of Functions (first part of this topic) should be function f (x) = 2x + 6 and of f -1, the Inverse function of f. (4). thoroughly mastered before this section on Inverse Functions . Determine the Inverse function f -1 in the form f -1 (x) = .. (2). In particular, work through Questions 2 7. Sketch the graphs of f and f -1 on the same set of axes, NB: The Inverse of a function REVERSES the process of indicating also the line y = x. (4) Sketch the graphs of the Functions f , f -1 and the line a function. y = x on the same set of axes. What do you notice? (4). Write down the equation of f -1 in the form f -1 (x) =.

3 (2). -1. Is f an increasing or a decreasing function? (1). y Are f and f -1 both Functions ? Why (not)? (2). f 1. The graph of f is .. 4. Given : g (x) = 3x - 2. Determine each of the following : Write down the coordinates of the reflections of the 1 following points in the line y = x : O x g - 1 (x) 1 g 1 (6).. -2 g ( x) x P(0; 0), Q(- 1;1), R(2; 4) and S(3; 9). x Let the images be P , Q , R and S respectively. (4). Sketch the graph y = 2 , indicating the coordinates of 2. If the Inverse of f is the reflection of f in the line y = x, Draw a sketch of the graph f which has equation y = x any three points on the graph. (3). then the graph of the Inverse is : for x m 0. (2). Use the three points on the sketch to write down the A. y B. y coordinates of three points on the Inverse function of Which of the points in question lie on the graph of f ? (2). x 2 y=2 . (3) f and its Inverse function, f -1, are reflections in the line.

4 ? (1). 1 O x 2 On the same system of axes, sketch the Inverse function O x x Draw the graph f -1, the Inverse function of f , on the -1 of y = 2 and the line y = x. (3) same system of axes. (2). x Describe the transformation from the graph y = 2 to its Which of the points in question ( question or answer). C. y D. y Inverse in words and give the rule for this transformation. (2) lie on the graph of f -1 ? (2). 2 Write down the equation of the Inverse function in the Complete : (a) f (3) = .. O x 1. form x = .. O x (b) f -1 (9) = .. -2 1 [ In Topic 4, you will convert this equation to y = .. ] (2). (c) f (2) = 4 ( ; ) lies on f (2) Are both the above graphs Functions ? Why (not)? (2). (d) f -1 (4) =2 ( ; ) lies on f -1 (4). Write down the domain and the range of the graphs of : Consider the graph of f given above. x y (a) y = 2 (b) x = 2 (4) Determine the equation of f -1 in the form f -1 (x) = .. (2).

5 Write down the equation of the given function and of the Inverse function in the form y = .. (2) 2 Is f -1 a function? Give a reason for your answer. (2). 6. Consider the function f where f (x) = 2x Hence complete: (a) f (x) = .. (b) f -1(x) =.. (2) 2. Write down the domain and range of f. (2) 9. If f (x) = (x + 2) ; x [ - 2, then f -1 (x) is equal to -1 2. Show how the equation of f could have been Sketch the graph of f and g on the same set of axes A. x - 2 B. x -2 C. x-2 D. - x -2 (2). calculated from the equation of f. (2) where g is the reflection of f in the line y = x. Draw the line y = x on the sketch. (3) 10. Given h - 1 (x) = - x . Then the equation of h is y = .. Explain why f and f -1 are both one-to-one 2 2 2. relations. (2) Determine the equation of g in the form y = .. (2) A. x ; x 0 B. x ; x 0 C. x D. x (2). Copyright The Answer 2. Gr 12 Maths Functions : Questions 11. Given : g (x) = -1 + x.]

6 NOTES. Determine the Inverse of g (x) in the form g -1 (x) = .. (4). y 12. f (x) = - 2(x - 3)(x + 1) and D. h (x) = m x + c h E S(2; a). The graphs of f and h have a common x-intercept at Q and f a common y-intercept at E. The turning point of f is at D O x Q. and S(2; a) is a point on f. Calculate the coordinates of E and D. (4). Write down the coordinates of Q. (1). Determine the numerical values of : (a) m (b) a (2)(2). Write down the coordinates of the turning point of f -1 , the Inverse of f. (2). Is f -1 a function? Why (not)? (2). 13. The graph of the function g is shown below. y 6- 4- g 2- O x -4 -2 2 4 6. -2 - -4 - Determine the domain and range of the function. (2). On this set of axes, draw the graph of the Inverse function of g. (4). Explain why this Inverse is a function. (1). * Now do Paper E1 Q10 in Section 2 of this book. * The Topic Guide indicates the examples in all the papers. * See the end of Topic 4 for mixed Questions (including exponential and log Functions ) on Inverse Functions .

7 Copyright The Answer 3. Gr 12 Maths Functions : Answers x NB : The Inverse of a function reverses the process of a function. x -3 0 3 y y=2. Inverse Functions So : f maps x = 1 onto y = 3. f -1 maps x = 3 onto y = 1. whereas & y=2. x 1. 8. 1 8. f y=x (3 ; 8). (Gr 12 only) (2; 0) lies on f and (0; 2) lies on f -1. (8 ; 3). x=2. y That is why x and y are swopped to determine the -3; 1 (0; 1). f -1. ANSWERS graph (and the equation) of the Inverse of a function. 8 . O x (1 ; 0). (8 ; 3), (1 ; 0) and 1 ; -3 . y . f: y = 2x + 6 1 8 . ; - 3 . 8 . 1. The given graph has points (- 2 ; 0) 1) x-int. : 2x + 6 = 0 f The graph of the Inverse has points (0 ; - 2) 0) 2x = - 6 y=x (put y = 0) (0 ; 6) f is reflected in the line y = x to produce the image f -1 . C x = -3 The rule : (x ; y) (y ; x ) . (- 3; 0) f -1. Given : y = 1 x + 1 C : y = 2x - 2 y 2 x = 2 .. In Topic 4, you will convert this to y = log 2 x, y-int. : y = 6 O x (3 ; 0) (6 ; 0) f -1 (x) = log 2 x, by using the definition of a log.

8 (a) f (x) = 1 x + 1 (b) f -1 (x) = 2x - 2 (put x = 0) (0; 6) . 2 (0 ; - 3). Yes ; for every x-value, there is only one y-value. Equation of f : y = 1x +1 [They are both one-to-one relations.]. 2. Equation of f -1 : x = 1y+1 .. swop x & y (a) Domain : x & Range : y > 0 ; y . 2. For f -1 (the Inverse function). % 2) 2x = y + 2 f and f -1. are reflections (b) Domain : x > 0 ; x & Range : y . x-int. : (6; 0) . y = 2x - 2 .. make y the subject .. in the line y = x. Note: The domains and ranges are swopped. y-int. : (0; - 3) So, swop x and y. For each value of x, there is only one y-value for both graphs. Equation of f -1 : y = mx + c where m = + 1 & c = - 3. 2 Domain : x ; Range : y m 0 ; y . Domain : x & Range : y . y = 1 x-3 by inspection on the sketch 2 y Equation of f : y = - 3x + 6. f -1 (x) =1 x-3 . Equation of f -1 : x = - 3y + 6 2 y=x f 3y = - x + 6 OR : Swop x and y in y = 2x + 6 and then make y the subject.

9 Y = -x +2. 3. Yes ; for both f and f -1 , each x-value is associated with f -1 (x) = - x +2 .. f (x) = - 1 x + 2 O x 3 3 only one y-value.. y Equation of g : y = 3x - 2 g f Equation of g - 1 : x = 3y - 2 2. 6 y=x Equation of f : y = 2x x + 2 = 3y We notice that f and f -1. Equation of g : x = 2y2. f -1 + 3) y = x + 2. are reflections in the 2. 2 3 3 2y = x x = x line y = x.. a straight line [ g (x) = 1 x + 2 ]. O. = x + 2 . -1 2. 2 6 g - 1 (x) y 3 3 3 3 2. y = x . 1 1 NB: f -1 (x) g 1. = .. a hyperbola ! 2. g (x) 3x - 2 f (x). f - 1 is the Inverse of f No ; g is the Inverse , but not the Inverse function, because 1 1. f (1) = 3 f -1 (3) = 1 g = 3 - 2 it is not a function. It is a one-to-many relation.. x x (not of f(x) ). 1. f (2) = 0 (2 ; 0) lies on f and (0 ; 2) lies on f -1 . g = 3 - 2 .. a hyperbola! Note: A vertical line could cut g more than once. x x Copyright The Answer A1. Gr 12 Maths Functions : Answers (a) See the 2 ways in (1) and (2) below.

10 Y P(0; 0), R(2; 4) Note : g: f (1) Consider y = 2x ; x [ 0. 2. (2) Consider y = 2x ; x m 0. 2 & 9) . S(3; 9) -1. y y y = x . f f f -1 Graph of y = x & Graph of y = - 1 + x R(2; 4). See sketch. S (9; 3). O O R (4; 2) g-1: x x P (0; 0), R (4; 2). 3) . 2 2. O x -1 -1. (b) Eqn. of f : y = 2x ; x [ 0 y = 2x ; x m 0 P(0 ; 0). - x f -1 (x) = + x 2 2. f -1 (x) = Graph of y = (x + 1) & Graph of y = (x + 1) ; x m - 1. 2 2. (a) f (3) = 9 (c) f (2) = 4 (2 ; 4) lies on f . y y y=x f y=x (b) f -1 (9) =3 (d) f - 1 (4) = 2 (4 ; 2) lies on f -1 At E, x = 0 At D, x = -1 + 3 .. ave. of 2 roots f f -1 2 & f (0) = - 2(- 3)(1) = 6 = 1. x Equation of f : y = x ; x m 0. O O x 2 E (0; 6) & f (1) = - 2(- 2)(2) = 8. Equation of f -1 : x = y f -1 y 2. = x Q (3; 0) .. f(3) = 0 D (1; 8) . y = x ; but y m 0. (c) For f -1 : For f -1 : (a) m = - 6 = - 2 . y = x .. y = + x 3.. Gradient of h Domain : x m 0 ; x Domain : x m 0 ; x . f -1 (x) = x.]]