QUESTION 1
(a) Let A = {1,2,3,4,5} and define $f: A \to \mathbb{Z}$ by
$f(x) = x^2 + 1$ if x is even, and $f(x) = 2x - 5$ if x is odd.
(i) Express $f$ as a subset of $A \times \mathbb{Z}$.
(ii) Is $f$ one-to-one? Give a reason for your answer.
(b) Find the largest subset A of $\mathbb{R}$ such that the given formula for $f(x)$ defines a function
with domain A. Give the range of $f$.
$\qquad f(x) = \frac{1}{\sqrt{1 - x}}$
QUESTION 2
Show that the following function is one-to-one. Find the range of $f$ and the inverse of $f$.
$f(x) = 5 - \frac{1}{1 + x}$
QUESTION 3
Let $S = \{1, 2, 3, 4, 5\}$ and let $f, g, h: S \to S$ be functions defined by
$f = \{(1, 2), (2, 1), (3, 4), (4, 5), (5, 3)\}$
$g = \{(1, 3), (2, 5), (3, 1), (4, 2), (5, 4)\}$
$h = \{(1, 2), (2, 2), (3, 4), (4, 3), (5, 1)\}$
(a) Explain why $f$ and $g$ have inverses but $h$ does not. Find $f^{-1}$ and $g^{-1}$.
(b) Show that $(f \circ g)^{-1} = g^{-1} \circ f^{-1} \ne f^{-1} \circ g^{-1}$.
