Sets and Functions Solved Examples (Class 11 Maths)
Sets and functions are fundamental concepts in mathematics. These examples cover set operations, Venn diagrams, domain and range of functions, and composit
TL;DR: Sets and functions are fundamental concepts in mathematics. These examples cover set operations, Venn diagrams, domain and range of functions, and com…
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Sets and functions are fundamental concepts in mathematics. These examples cover set operations, Venn diagrams, domain and range of functions, and composit
Sets and Functions — Solved Numerical Examples (Step by Step)
Example 1: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find (a) A ∪ B (b) A ∩ B (c) A - B
Solution: (a) Union (A ∪ B) includes all elements in A or B or both:
A ∪ B = {1, 2, 3, 4, 5, 6}
(b) Intersection (A ∩ B) includes elements common to both A and B:
A ∩ B = {3, 4}
(c) Difference (A - B) includes elements in A but not in B:
A - B = {1, 2}
Example 2: If n(A) = 15, n(B) = 20, and n(A ∩ B) = 8, find n(A ∪ B).
Solution: Using the formula for union of sets:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
n(A ∪ B) = 15 + 20 - 8
n(A ∪ B) = 27
Example 3: Find the domain and range of the function f(x) = sqrt(x - 2).
Solution: For the function f(x) = sqrt(x - 2) to be defined, the expression under the square root must be non-negative:
x - 2 ≥ 0
x ≥ 2
Domain = {x | x ≥ 2} or [2, ∞)
Since sqrt(x - 2) ≥ 0 for all x in the domain:
Range = {y | y ≥ 0} or [0, ∞)
Example 4: If f(x) = 2x + 3 and g(x) = x² - 1, find (a) (f ∘ g)(x) (b) (g ∘ f)(x).
Solution: (a) (f ∘ g)(x) = f(g(x)) = f(x² - 1)
= 2(x² - 1) + 3
= 2x² - 2 + 3
= 2x² + 1
(b) (g ∘ f)(x) = g(f(x)) = g(2x + 3)
= (2x + 3)² - 1
= 4x² + 12x + 9 - 1
= 4x² + 12x + 8
Example 5: Determine whether the function f(x) = x³ is one-to-one (injective).
Solution: A function is one-to-one if f(a) = f(b) implies a = b.
Suppose f(a) = f(b):
a³ = b³
Taking cube root of both sides:
a = b
Therefore, f is one-to-one (injective).
Alternatively, the horizontal line test shows that any horizontal line intersects the curve y = x³ at most once.
Example 6: If f(x) = 2x - 5, find the inverse function f⁻¹(x).
Solution: Let y = f(x) = 2x - 5
To find the inverse, swap x and y, then solve for y:
x = 2y - 5
2y = x + 5
y = (x + 5) / 2
Therefore, f⁻¹(x) = (x + 5) / 2
Verification: f(f⁻¹(x)) = f((x + 5)/2) = 2((x + 5)/2) - 5 = x + 5 - 5 = x ✓
Tips
- Use Venn diagrams to visualize set operations and solve problems involving overlapping sets.
- For composite functions, remember that (f ∘ g)(x) means apply g first, then apply f to the result.
- Domain consists of all valid input values (x), while range consists of all possible output values (y).
- A function has an inverse only if it is both one-to-one (injective) and onto (surjective), making it bijective.
Frequently Asked Questions
What is the difference between a relation and a function?
A relation is any set of ordered pairs. A function is a special relation where each input (x-value) corresponds to exactly one output (y-value). Every function is a relation, but not every relation is a function.
Can a function have the same range for different domains?
Yes, different functions can have the same range. For example, f(x) = x² defined on [-2, 2] and g(x) = x² defined on [0, 3] both have ranges that include certain values, though the specific ranges differ.
More Maths Solved Examples
- Quadratic Equations
- Trigonometry
- Arithmetic Progressions
- Surface Areas and Volumes
- Coordinate Geometry
- Probability
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