**Problem 1**. Solve the following functions for x (where x is a real number). Leave your answers in exact form, that is, do not use a calculator, show all working.

**Problem 2**. A manager has determined the cost C, in dollars, for manufacturing is

for manufacturing, where q is the number of units produced in a given day.

(a) What is the cost of manufacturing before any units are produced?

(b) What is the cost of manufacturing if 1980 units are produced in a given day?

(c) How many units would be produced if the cost of manufacturing is $150 in a given day?

**Problem 3**. Find all solutions of the following equations in the interval [0, 2π)

**Problem 4**. (a) Sketch the graph of the circular function

i. State clearly the amplitude, phase shift, period and the equation of the midline.

ii. Show working for all θ- and y-intercepts for the function y = f(θ).

iii. Show at least two periods for the graph of y = f(θ), and label the axes clearly.

(b) Prove the following identity

Make sure you show all working and give clear explanations.

**Problem 5**. Consider the function

which has the graph

(a) Explain why f has no inverse function. You should include an example to support your explanation.

(b) Determine the largest possible domain, which includes x = 1, for f(x) such that the inverse function f^{-1}(x) does exist.

(c) Given your answers in (b) ?nd the inverse function f^{-1}(x). Clearly explain key steps and show all working.

(d) Sketch the graph of y = f(x) for the restricted domain in (b) and y = f-1(x) on the same set of axes. All points of intersection and axes-intercepts should be easily determined from your sketch or clearly labelled. Furthermore axes should be clearly labelled with appropriate scaling.

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