Domain

Exercises

1. Find the domain of the following functions, compute $f(0)$, $f(-1)$; establish whether they are even, odd or not: a) $f_1(x) = \frac{x-3}{x-2}$ b) $f_2(x) = |x-5|$ c) $f_3(x) = -7$ d) $f_4(x) = \left|\frac{x-3}{x-2}\right|$ e) $f_5(x) = \frac{x+1}{x^2-3x+2}$

2. Find the domain of the following functions, establish if they are monotone and/or bounded. Draw their graphs: a) $f(x) = \begin{cases} -3 & x < 0 \ 3 & x \geq 0 \end{cases}$ b) $f(x) = \begin{cases} -1 & x < 0 \ -x & x \geq 0 \end{cases}$ c) $f(x) = \begin{cases} -x & -3 < x < -1 \ 1 & -1 \leq x < 2 \ x + 1 & 2 \leq x \end{cases}$

3. Determine the following compositions (and the domains): $f_1 \circ f_2$ and $f_2 \circ f_1$, where $f_1(x) = \frac{x-3}{x-2}$ and $f_2(x) =

   x-5

   $

4. Find the domain of the following functions: a) $\sqrt{\frac{2x-1}{x+2}}$ b) $s^{1/3} - s^{1/2}$ c) $\sqrt[3]{1 - \sqrt[3]{x^2 - 4}}$ d) $\sqrt{|x-3| - |x+4|}$ e) $\sqrt{\frac{x-1}{2|x|+3}}$ f) $(x-1)^{\sqrt{3-2x}}$ g) $\sqrt[4]{\lg(3x^2+2x)}$ h) $\frac{x}{e-e^{1/x}}$ i) $\lg \frac{t-2}{t}$ j) $x \lg(x^2-3)$ k) $\ln\ln x$ l) $\sqrt{\frac{\lg x-1}{x}}$ m) $\sin \frac{t}{t+1}$ n) $\frac{\sin x}{(1+\cos x)^2}$ o) $\ln(1-\arctan x)$ p) $\arcsin(1+x)$ q) $\arccos \frac{x}{x+1}$ r) $\sqrt{\ln(\sin t-\cos t)}$, for $t \in [0,2\pi]$ s) $\frac{1}{x}\lg\frac{e^x-1}{x}$ t) $\sqrt{2-\sqrt{\frac{t-9}{t-1}}}$

5. Establish if the following functions are odd, even or neither odd nor even: a) $f(t) = \lg(t^2-1)$ b) $f(x) = x\cos x$ c) $f(x) = e^{x^3}$