Functions

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: ...