Question

1. (6 marks) Provide the domain, target, and range of the following functions (a, b, c, d}3 . For each x E(a, b, cF, fx)-dx. a) fta, b. c}2 b) g: {a, b, c, d)-(a, b, c, d}?. For each x E(a, b, c, d], g(x) (4 marks) Use the ceiling and floor functions to qive a mathematical expression for the following a) Among a random group of 100 people at least 9 must be born in the same month b) Knowing your sock drawer has 3 unique pairs of socks. If you couldn't see the socks how xbx 2. many would you have to pick before you guaranteed you were holding at least a matching ail (4 marks) Are the following functions from R to R? If not, explain why. a) f(x) 1/x b) f(x)- Vx 3. (9 marks) Foreach of the functions below (from Z to Z), indicate whether the function is onto, one-to-one, neither or both. If the function is not onto or not one-to-one, give an example showing why. If the function is bijective, find and show its inverse 4. a) f(x) -x-1 b) j(x) -x 1 c) f(x)

Answer 1

SOLUTION TO QUESTION 1:

a) $f: \{a,b,c\}^2 \rightarrow \{a,b,c,d\}^2$ For each $x \in \{a,b,c\}^2$ , $g(x)=dx$

$g(x)= d \times \{a,b,c\}^2 \\ ~~~~~~~~~~=d \times \{a,b,c\} \times \{a,b,c\} \\ ~~~~~~~~~~~=\{daa,dab,dac,dad,dba,dbb,dbc,dbd,dca,dcb,dcc\}\\ ~~~~~~~~~~=d \times \{a,b,c\}^2$

The domain of a function is the set of all inputs that produce an output. So, the domain of f is  $\{a,b,c\}^2$.

The range of a function is the set of all outputs obtained f(x). So, the range of the function is  $d\{a,b,c\}^2$ .

The target of a function is the set of all possible elements in output space. So, the target of the function is  $\{a,b,c\}^3$ .

b) $g: \{a,b,c,d\} \rightarrow \{a,b,c,d\}^3$ . For each $x \in \{a,b,c,d\}$ , $g(x)=xbx$

$g(x)= \{a,b,c,d\} \times b \times \{a,b,c,d\} \\ ~~~~~~~~~~~=\{aba,abb,abc,abd,bba,bbb,bbc,bbd,cba,cbb,cbc,cbd,dba,dbb,dbc,dbd\}\\ ~~~~~~~~=\{a,b,c,d\} \times d \times \{a,b,c,d\}$

The domain of g is  $\{a,b,c,d\}.$ .

The range of g is $\{a,b,c,d\} \times d \times \{a,b,c,d\}$ .

The target function of g is $\{a,b,c,d\} \times \{a,b,c,d\} \times \{a,b,c,d\}=\{a,b,c,d\}^3$