2.6 Miscellaneous Exercise on Chapter 2

Question 1.


The relation f is defined by f(x) = {x^2, 0 ≤ x ≤ 3; 3x, 3 ≤ x ≤ 10


The relation g is defined by g(x) = {x^2, 0 ≤ x ≤ 2; 3x, 2 ≤ x ≤ 10


Show that f is a function and g is not a function.

Question 2.


If f (x) = x^2, find [f(1.1) − f(1)]/(1.1 − 1).

Answer 2.


2.1

Question 3.


Find the domain of the function f (x) = (x^2 + 2x + 1)/(x^2 − 8x + 12)

Answer 3.


Domain of function is set of real numbers except 6 and 2.

Question 4.


Find the domain and the range of the real function f defined by f (x) = √ (x − 1).

Answer 4.


Domain = [1, ∞),


Range = [0, ∞)

Question 5.


Find the domain and the range of the real function f defined by f (x) = |x −1|.

Answer 5.


Domain = R,


Range = non-negative real numbers

Question 6.


Let f {(x, x^2/[1 + x^2]) : x ∈ R be a function from R into R. Determine the range


of f.

Answer 6.


Range = [0, 1)

Question 7.


Let f, g : R → R be defined, respectively by f(x) = x + 1, g(x) = 2x − 3. Find f + g, f − g and f/g.

Answer 7.


(f + g) x = 3x − 2


(f − g) x = − x + 4


(f/g)x = x + 1/2x − 3, x ≠ 3/2

Question 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.

Answer 8.


a = 2, b = − 1

Question 9.


Let R be a relation from N to N defined by R = {(a, b) : a, b ∈ N and a = b^2}. Are the following true?


(i) (a, a) ∈ R, for all a ∈ N


(ii) (a, b) ∈ R, implies (b, a) ∈ R


(iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.


Justify your answer in each case.

Answer 9.


(i) No


(ii) No


(iii) No

Question 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.

Answer 10.


(i) Yes,


(ii) No

Question 11.


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

Answer 11.


No

Question 12.


Let A = {9,10,11,12,13} and let f : A → N be defined by f (n) = the highest prime factor of n. Find the range of f.

Answer 12.


Range of f = {3, 5, 11, 13 }