Functions

Exponential, Logarithmic and Hyperbolic Functions Exercise 1 : Basic Exponent Rules Calculate the terms given in the next questions or give a transformation: $a^{-n}$ $27^{\frac{1}{3}}$ $a^{\frac{1}{n}}$ $(0,1)^0$ $(y^3)^2$ $x^{-\frac{3}{2}}$ $10^3 \cdot 10^{-3} \cdot 10^2$ $3^{-3}$ Exercise 2 : Roots and Powers Calculate the terms given in the next questions or give a transformation: $(\sqrt{2})^2$ $e^{\frac{1}{10}}$ $(\ln 2)^0$ $\sqrt{5} \cdot \sqrt{7}$ $(0,5)^2 \cdot (0,5)^{-4} \cdot (0,5)^0$ $\sqrt{8} \cdot \sqrt{3}$ Exercise 3 : Common Logarithms (base 10) Calculate the terms given in the next questions or give a transformation: ...

Dirac Delta Function Exercise 1 : Basic Delta Integrals Evaluate the following integrals: $\displaystyle \int_{2}^{6} (3x^2 - 2x - 1) , \delta(x - 3) , dx$ $\displaystyle \int_{0}^{5} \cos x ; \delta(x - \pi) , dx$ $\displaystyle \int_{-1}^{3} x^3 , \delta(x + 1) , dx$ $\displaystyle \int_{-\infty}^{\infty} \ln(x + 3) , \delta(x + 2) , dx$ $\displaystyle \int_{0}^{\infty} e^{-x} , \delta(x - 2) , dx$ Exercise 2 : Delta with Scaled & Shifted Arguments Use $\delta(ax) = \frac{1}{|a|}\delta(x)$ and similar properties to evaluate: ...

Inverse Functions Exercise 1 Find $ f^{-1} $ for each function $ f $ $ f(x) = x^3 + 1 $ $ f(x) = (x - 1)^3 $ $ f(x) = \begin{cases} x & \text{rational} \ -x & \text{irrational} \end{cases} $ $ f(x) = \begin{cases} -x^2 & x \geq 0 \ 1 - x^3 & x < 0 \end{cases} $ $ f(x) = \begin{cases} x & x \neq a_i \ a_{i+1} & x = a_i \ (i < n) \ a_1 & x = a_n \end{cases} $ $ f(x) = x + \lfloor x \rfloor $ $ f(0.a_1a_2a_3\ldots) = 0.a_2a_1a_3\ldots $ $ f(x) = \frac{x}{1 - x^2}, \ -1 < x < 1 $ Exercise 2 Describe $ f^{-1} $’s graph when $ f $ is: ...

Functions Exercise 1 Let $f(x) = \frac{1}{1 + x}$. Find: $f(f(x))$ and determine its domain $f\left(\frac{1}{x}\right)$ $f(cx)$ $f(x + y)$ $f(x) + f(y)$ For which numbers $c$ does there exist $x$ such that $f(cx) = f(x)$? For which numbers $c$ does $f(cx) = f(x)$ hold for two different values of $x$? Exercise 2 Let $g(x) = x^2$ and $h(x) = \begin{cases} 0 & \text{if } x \text{ rational} \ 1 & \text{if } x \text{ irrational} \end{cases}$. Find: ...