# 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)**