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 which
b) On your sketch show two of the lines of symmetry which the complete graph possesses.
(Marks available: 4)
Answer outline and marking scheme for question: 1
Give yourself marks for mentioning any of the points below:
a) The graph would look like:
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
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 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)
(3 marks)
(Marks available: 7)
3. The functions f and g are defined for all real numbers by:
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
(Marks available: 10)
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:
(ii) Solving g (x) =1/2, gives the same results as above, but with ±.
(6 marks)
(Marks available: 10)
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