Programa Diploma BI. Funciones Given that f(x) = 2e 3x, find the inverse function f 1 (x). Working: Answers:

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1 Funciones. Given that f() = e, find the inverse function f ().... The functions f() and g() are given b f() = and g() = +. The function (f g)() is defined for, ecept for the interval ] a, b [. (a) Calculate the value of a and of b. (b) Find the range of f g. (Total 6 marks)

2 Funciones. Consider the functions f : a 4 ( ) and g : a (a) Find g. 6. (b) Solve the equation (f g ) () = 4. (Total 6 marks) 4. Let f() =, and g() =, ( ). Find (a) (g f) (); (b) g (5). (Total 6 marks)

3 Funciones 5. Consider the function f : a +, (a) Determine the inverse function f. (b) What is the domain of f? 6. The function f is defined b f : a,. Evaluate f (5)...

4 4 Funciones 7. The diagram shows a sketch of part of the graph of f() = and a sketch of part of the graph of g() = + 6 =( f ) =g() (a) Write down the coordinates of the maimum point of = g(). The graph of = g() can be obtained from the graph of = f() b first reflecting the graph of = f(), then translating the graph of = f(). (b) Describe full each of these transformations, which together map the graph of = f() onto the graph of = g(). 4

5 5 Funciones... (Total marks) 5

6 6 Funciones 8. The diagram shows the graph of = f(), with the -ais as an asmptote. B(5, 4) A( 5, 4) (a) On the same aes, draw the graph of =f( + ), indicating the coordinates of the images of the points A and B. (b) Write down the equation of the asmptote to the graph of = f( + ). (b).. 6

7 7 Funciones 9. A group of ten leopards is introduced into a game park. After t ears the number of leopards, N, is modelled b N = 0 e 0.4t. (a) (b) How man leopards are there after ears? How long will it take for the number of leopards to reach 00? Give our answers to an appropriate degree of accurac. Give our answers to an appropriate degree of accurac. 7

8 8 Funciones 0. The following graph is that part of the graph of = f() for which f() Sketch, on the aes provided below, the graph of = f() for (Total marks) 8

9 9 Funciones. The diagrams show how the graph of = is transformed to the graph of = f() in three steps. For each diagram give the equation of the curve. (a) 0 = 0 (b) (c) (c)... 9

10 0 Funciones. Given functions f : a + and g : a, find the function (f g) l.... (Total marks). The diagram represents the graph of the function f : a ( p) ( q). C (a) Write down the values of p and q. (b) The function has a minimum value at the point C. Find the -coordinate of C. 0

11 Funciones 4. Three of the following diagrams I, II, III, IV represent the graphs of (a) = + cos (b) = cos( + ) (c) = cos +. Identif which diagram represents which graph. I II 4 π π III IV 5 4 π π (c)...

12 Funciones 5. The function f is given b f() = n ( ). Find the domain of the function A population of bacteria is growing at the rate of. % per minute. How long will it take for the size of the population to double? Give our answer to the nearest minute

13 Funciones 7. For which values of the real number is + k = + k, where k is a positive real number? The diagram shows three graphs. B A C A is part of the graph of =. B is part of the graph of =. C is the reflection of graph B in line A. Write down (a) the equation of C in the form =f(); (b) the coordinates of the point where C cuts the -ais.

14 4 Funciones 9. Two functions f, g are defined as follows: Find (a) f (); f : + 5 g : ( ) (b) (g o f)( 4). 4

15 5 Funciones 0. Let f() =, and g() =. Solve the equation (f o g)() =