Answers to exercises
Exercise 1
 Are 'diagonalise' and 'diagonalize' two words or the same word?
 Is the phrase 'deja vu' two English words? What about 'café'?
 Can we restrict English words to the words defined in the Oxford English Dictionary? Is the word 'gunna' defined? Was 'internet' in the Oxford English Dictionary in 1970?
Exercise 2
Let \(f(x) = \dfrac{x^2}{2} + x\).
 \(f(3) = \dfrac{3^2}{2} + 3 = \dfrac{15}{2}\).
 \(f(a+h) = \dfrac{(a+h)^2}{2} + a + h\).
Exercise 3
 Let \(f(x) = \sqrt{(x1)(x2)}\). Then \(f(x)\) is defined if and only if \((x1)(x2) \geq 0\). So we sketch the graph of \(y=(x1)(x2)\).
Hence, \(\mathrm{domain}(f) = (\infty,1] \cup [2,\infty)\) and \(\mathrm{range}(f)=[0,\infty)\).  Let \(g(x) = \sqrt{12\sin2x}\). To find the domain of \(g(x)\), we first note that
\begin{align*}
g(x) \text{ is defined} \ &\iff\ 12\sin2x \geq 0 \\
&\iff\ 2\sin2x \leq 1 \\
&\iff\ \sin2x \leq \tfrac{1}{2}.
\end{align*}
So we sketch the graph of \(y = \sin 2x\).
Now solve \(\sin2x = \dfrac{1}{2}\) for \(x\): \begin{alignat*}{4} 2x &= \dfrac{\pi}{6} + 2k\pi &\quad&\text{or}\quad & 2x &= \dfrac{5\pi}{6} + 2k\pi, &\quad&\text{for some } k \in \mathbb{Z} \\ x &= \dfrac{\pi}{12} + k\pi &&\text{or} & x &= \dfrac{5\pi}{12} + k\pi, &&\text{for some } k \in \mathbb{Z}. \end{alignat*} It follows that \(\sin2x \leq \dfrac{1}{2}\) if and only if \(x\) belongs to the interval \(\Bigl[\dfrac{5\pi}{12}, \dfrac{13\pi}{12}\Bigr]\) or one of its translates by a multiple of \(\pi\). Hence, \[ \mathrm{domain}(g) = \bigcup_{k \in \mathbb{Z}}\, \Bigl[\dfrac{5\pi}{12} + k\pi, \dfrac{13\pi}{12}+k\pi\Bigr]. \] The maximum value of \(g(x)\) is \(\sqrt{1  2\times (1)} = \sqrt{3}\). Hence, \(\mathrm{range}(g) = [0,\sqrt{3}]\).
Exercise 4
Let \(f(x)=3\tan 2x\). The domain of \(\tan x\) is all real numbers except odd multiples of \(\dfrac{\pi}{2}\), and the range of \(\tan x\) is all reals. Hence, \[ \mathrm{domain}(f) = \mathbb{R} \setminus \{\dots, \tfrac{3\pi}{4}, \tfrac{\pi}{4}, \tfrac{\pi}{4}, \tfrac{3\pi}{4}, \dots\} \] and \(\mathrm{range}(f) = \mathbb{R}\).Exercise 5

\(\mathrm{domain}(f) = \mathbb{R}\) and \(\mathrm{range}(f) = \mathbb{Z}\). 
\(\mathrm{domain}(g) = \mathbb{R}\) and \(\mathrm{range}(g) = [0,1)\).
Exercise 6
 Arithmetic sequence with first term \(a=2\) and common difference \(d=3\).
 Geometric sequence with first term \(a=3\) and common ratio \(r=2\).
 Geometric sequence with first term \(a=1\) and common ratio \(r=0.1\).
 The Fibonacci sequence; it is neither geometric nor arithmetic.
 This sequence is neither geometric nor arithmetic.
Exercise 7
\(\mathrm{domain}(f) = \mathrm{codomain}(f) = \mathbb{N}\) and \(\mathrm{range}(f) = \{3,5,7,9,11,\dots\}\).
Exercise 8
Assume \(A = \{a_1,a_2,\dots,a_m\}\) and \(B = \{b_1,b_2,\dots,b_n\}\). If we are defining a function from \(A\) to \(B\), then there are \(n\) choices for where to map \(a_1\), and then there are \(n\) choices for where to map \(a_2\), and so on. Thus the total number of functions is \(n^m\).