Advertisements
Advertisements
Question
Find the domain of the real valued function of real variable:
(i) \[f\left( x \right) = \frac{1}{x}\]
Solution
(i) Given:
We observe that f (x) is defined for all x except at x = 0.
At x = 0, f (x) takes the intermediate form \[\frac{1}{0} .\]
APPEARS IN
RELATED QUESTIONS
State whether the following statement is true or false. If the statement is false, rewrite the given statement correctly.
If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B.
If A × B = {(a, x), (a, y), (b, x), (b, y)}. Find A and B.
Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that A × (B ∩ C) = (A × B) ∩ (A × C)
Let A and B be two sets such that n(A) = 3 and n (B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements.
If A = {1, 2, 3} and B = {2, 4}, what are A × B, B × A, A × A, B × B and (A × B) ∩ (B × A)?
Let A = {1, 2, 3, 4} and R = {(a, b) : a ∈ A, b ∈ A, a divides b}. Write R explicitly.
State whether of the statement is true or false. If the statement is false, re-write the given statement correctly:
If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}
State whether of the statement is true or false. If the statement is false, re-write the given statement correctly:
(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ ϕ) = ϕ.
If A = {1, 2}, from the set A × A × A.
Given A = {1, 2, 3}, B = {3, 4}, C ={4, 5, 6}, find (A × B) ∩ (B × C ).
If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
(i) A × (B ∪ C) = (A × B) ∪ (A × C)
If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
(iii) A × (B − C) = (A × B) − (A × C)
Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that:
(i) A × C ⊂ B × D
If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find
(i) A × (B ∩ C)
If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find
(iv) (A × B) ∪ (A × C)
Find the domain of the real valued function of real variable:
(iv) \[f\left( x \right) = \frac{2x + 1}{x^2 - 9}\]
Find the domain of the real valued function of real variable:
(v) \[f\left( x \right) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}\]
Find the domain of the real valued function of real variable:
(i) \[f\left( x \right) = \sqrt{x - 2}\]
Find the domain of the real valued function of real variable:
(ii) \[f\left( x \right) = \frac{1}{\sqrt{x^2 - 1}}\]
Find the domain and range of the real valued function:
(i) \[f\left( x \right) = \frac{ax + b}{bx - a}\]
Find the domain and range of the real valued function:
(iii) \[f\left( x \right) = \sqrt{x - 1}\]
Find the domain and range of the real valued function:
(vi) \[f\left( x \right) = \left| x - 1 \right|\]
Find f + g, f − g, cf (c ∈ R, c ≠ 0), fg, \[\frac{1}{f}\text{ and } \frac{f}{g}\] in :
(a) If f(x) = x3 + 1 and g(x) = x + 1
Find f + g, f − g, cf (c ∈ R, c ≠ 0), fg, \[\frac{1}{f}\text{ and } \frac{f}{g}\] in :
(b) If \[f\left( x \right) = \sqrt{x - 1}\] and \[g\left( x \right) = \sqrt{x + 1}\]
Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine B × A
Let A = {–1, 2, 3} and B = {1, 3}. Determine A × B
Let A = {–1, 2, 3} and B = {1, 3}. Determine B × A
If P = {x : x < 3, x ∈ N}, Q = {x : x ≤ 2, x ∈ W}. Find (P ∪ Q) × (P ∩ Q), where W is the set of whole numbers.
State True or False for the following statement.
If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, then (A × B) ∪ (A × C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}.
State True or False for the following statement.
If A × B = {(a, x), (a, y), (b, x), (b, y)}, then A = {a, b}, B = {x, y}
