1. The relation f is defined by
f(x)=\{\begin{matrix}x^2,\mathrm{\ 0}\ \le x\le3\\3x,\mathrm{\ 3}\le x\le\mathrm{10}\\\end{matrix}
g(x)=\{\begin{matrix}x^2,\mathrm{\ 0}\ \le x\le2\\3x,\mathrm{\ 2}\le x\le\mathrm{10}\\\end{matrix}
Show that f is a function and g is not a function.
2. If f(x)=x^2, find \frac{f(1.1)-f(1)}{(1.1-1)}.
3. Find the domain of the function f\left(x\right)=\frac{x^2+2x+1}{x^2-8x+12}.
4. Find the domain and the range of the real function f defined by f\left(x\right)=\sqrt{x-1}.
5. Find the domain and the range of the real function f defined by f(x)=\left|x-1\right|.
9. Let R be a relation from N to N defined by R={(a,b) : a, b\in N and a=b^2}. Are the following true?
(i) (a,a) \in R, for all a\in N
(ii) (a,b) \in R, implies (b,a) \in R
(iii) (a,b) \in R, (b,c) \in R implies (a,c) \in R.
Justify your answer in each case.
10. Let A={1,2,3,4}, B={1,5,9,11,15,16} and f={(1,5),(2,9),(3,1),(4,5),(2,11)}
(i) f is a relation from A to B
(ii) f is a function from A to B.
Justify your answer in each case.
12. Let A={9,10,11,12,13} and let f : A \rightarrow N be defined by f(n)=the highest prime factor of n. Find the range of f.