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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}.

Relations and Functions

Miscellaneous Exercise

ncert-solutions-class-11-chapter-2-relations-and-functions-miscellaneous-exercise-question-7

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.