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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.

Relations and Functions

Miscellaneous Exercise

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Class 11

Relations and Functions

Miscellaneous Exercise

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 \(f(x)=\left|x-1\right|\).

6. Let \(f=\left\{\left(x,\frac{x^2}{1+x^2}\right):\mathrm{\ x} \in R\right\}\) be a function from R into R. Determine the range of f.

7. Let \(f, g : R\rightarrow R \) be defined, respectively by f(x)=x+1, g(x)=2x–3. Find f+g, f–g and \(\frac{f}{g}\).

8. Let f={(1,1),(2,3),(0,–1),(–1,–3)} be a function from Z to Z defined by \(f(x)=ax+b\), for some integers a, b. Determine a, b.

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)} 

Are the following true?

(i) f is a relation from A to B

(ii) f is a function from A to B.
Justify your answer in each case.

11. Let f be the subset of Z×Z defined by f={(ab,a+b) : a, b\(\in\) Z}. Is f a function from Z to Z? Justify your answer.

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.