Exam-style Questions: Functions

1. A graph has equation

y = cos2x,

where x is a real number.

a) Draw a sketch of that part of the graph for whichCopyright S-cool

b) On your sketch show two of the lines of symmetry which the complete graph possesses.

(Marks available: 4)

Answer

Answer outline and marking scheme for question: 1

Give yourself marks for mentioning any of the points below:

a) The graph would look like:

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Note 1: it must be a sine-wave shape - not W shape.

(the curve is only shown in the domain)

Note 2: Stationary Points at (0.1). (π/2.-1) .etc (degrees not allowed here)

(2 marks)

b) Lines of symmetry are:

x = 0 or π/2 or π etc

(you will get a mark for each correct line of symmetry, up to 2 marks).

(2 marks)

(Marks available: 4)

2. The function f is defined on the domain x > -1 by

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a) Write down the equations of the asymptotes to the curve y = f(x).

b) Give the range of the function f.

c) Give the domain and range of the inverse function f -1.

d) Find an expression for f -1(x).

(Marks available: 7)

Answer

Answer outline and marking scheme for question: 2

Give yourself marks for mentioning any of the points below:

a) Asymptotes are defined by the lines

x = -1, y = 2

(2 marks)

b) The range of f is y > 2

(1 mark)

c) The domain of f -1 is x > 2

The range of f -1 is x > -1

(1 mark)

d) Changing subject of y = f(x)

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(3 marks)

(Marks available: 7)

3. The functions f and g are defined for all real numbers by:

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a) (i) State whether f is an odd function, an even function or neither.

(ii) State whether g is an odd function, an even function or neither.

b) Given that f and g are periodic functions, write down the periods of f and of g.

c) Solve, for -π

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(Marks available: 10)

Answer

Answer outline and marking scheme for question: 3

Give yourself marks for mentioning any of the points below:

a) f is odd

g is even

(2 marks)

b) Period of f is π

Period of g is 1/2 π

(2 marks)

c) (i) Solving f (x) =1/2, gives:

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(ii) Solving g (x) =1/2, gives the same results as above, but with ±.

(6 marks)

(Marks available: 10)