Domain :
Let y = f(x) be a function.
Domain is all real values of x for which y is defined.
If there is any value of x for which y is undefined, we have to exclude that particular value from the set of domain.
Range :
Let y = f(x) be a function.
Range is all real values of y for the given domain (real values of x).
Let us look at some practice questions to understand how to find domain and range of a function.
Question 1 :
Find the domain of 1 / (1 − 2sinx)
Solution :
1 − 2sin x = 0
- 2sin x = - 1
sin x = 1/2
sin x = sin π/6
Since it sin function, the domain will be R - {nπ + (-1)n π/6}, n ∈ Z
Question 2 :
Find the largest possible domain of the real valued function f(x) = √(4 - x2)/ √(x2 - 9)
Solution :
Let us equate numerator and denominator equal to 0.
(4 - x2) = 0 x2 = 4 x = √4 x = ± 2 |
(x2 - 9) = 0 x2 = 9 x = √9 x = ± 3 |
(-∞, -3) (-3, -2) (-2, 2) (2, 3) (3, ∞)
If x ∈ (-∞, -3)
f(-3.5) = √(4 - (-3.5)2)/ √((-3.5)2 - 9)
= √(4 -12.25)/ √(12.25 - 9)
= √(-8.25)/ √3.25
= Not defined
Hence x ∉ (-∞, -3)
If x ∈ (-3, -2)
f(-2.5) = √(4 - (-2.5)2)/ √((-2.5)2 - 9)
= √(4 -6.25)/ √(6.25- 9)
= Not defined
Hence x ∉ (-3, -2)
If x ∈ (-2, 2)
f(0) = √(4 - 02)/ √((0)2 - 9)
= √4/ √(-9)
= Not defined
Hence x ∉ (-2, 2)
If x ∈ (2, 3)
f(2.5) = √(4 - (2.5)2)/ √((2.5)2 - 9)
= √(4 - 6.25)/ √(6.25-9)
= Not defined
Hence x ∉ (2, 3)
If x ∈ (3, ∞)
f(4) = √(4 - 42)/ √(4)2 - 9)
= √(4 - 16)/ √(16-9
= Not defined
Hence x ∉ (3, ∞).
Hence the answer is null set.
Question 3 :
Find the range of the function
1 / (2 cos x − 1)
Solution :
Range for cos function is between -1 and 1
-1 ≤ cos x ≤ 1
-2 ≤ 2cos x ≤ 2
-2 - 1 ≤ 2cos x - 1 ≤ 2 - 1
-3 ≤ 2cos x - 1 ≤ 1
Take reciprocal through out the equation, we get
-1/3 ≤ 1/(2cos x - 1) ≤ 1/1
-1/3 ≤ 1/(2cos x - 1) ≤ 1
(-∞, -1/3] U [1, ∞) is the required range.
Question 4 :
Show that the relation xy = −2 is a function for a suitable domain. Find the domain and the range of the function.
Solution :
xy = −2
y = -2/x
Domain means set of possible values of x.
Domain is all real values expect 0.
Domain = R - {0}
x = -2/y
Range means set of possible values of y.
Range is all real values expect 0.
Range = R - {0}
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