✨ Math Magic: Fun with Functions! ✨
Let f = {(x, y) | y ∈ N and 2y = x} be a relation on N. Find the domain, co-domain and range. Is this relation a function?
Step 1: Understand the Relation
It pairs x with y such that 2y = x. Example: If y = 1, then x = 2 → (2,1) is in f.
Step 2: Find Domain
All possible x-values where x = 2y and y ∈ ℕ → {2, 4, 6, 8, ...}
Step 3: Find Codomain
Given as ℕ (natural numbers)
Step 4: Find Range
All possible y-values where y = x/2 → ℕ (all natural numbers)
✅ Final Answer:
Domain = {2, 4, 6, ...}, Codomain = ℕ, Range = ℕ
Yes, it's a function!
Let X = {3, 4, 6, 8}. Determine whether the relation R = {(x, f(x)) | x ∈ X, f(x) = x² + 1} is a function from X to N?
Step 1: Compute f(x) for each x ∈ X
- f(3) = 3² + 1 = 10
- f(4) = 4² + 1 = 17
- f(6) = 6² + 1 = 37
- f(8) = 8² + 1 = 65
Step 2: Check Function Definition
Each x has exactly one f(x) value → satisfies function definition
✅ Final Answer:
Yes, R is a function!
Given the function f(x) = x² - 5x + 6, evaluate:
(i) f(-1) (ii) f(a²) (iii) f(2) (iv) f(x-1)
(i) f(-1)
= (-1)² - 5(-1) + 6 = 1 + 5 + 6 = 12
(ii) f(a²)
= (a²)² - 5(a²) + 6 = a⁴ - 5a² + 6
(iii) f(2)
= 2² - 5(2) + 6 = 4 - 10 + 6 = 0
(iv) f(x-1)
= (x-1)² - 5(x-1) + 6 = x² - 2x + 1 - 5x + 5 + 6 = x² - 7x + 12
A graph representing the function f(x) is given (imagine Fig.1.16). We know f(9) = 2.
(i) Find: (a) f(0) (b) f(7) (c) f(2) (d) f(10)
(ii) For what x is f(x) = 1?
(iii) Describe Domain and Range
(iv) What is the image of 6 under f?
Step 1: Read values from graph
(Would need actual graph to determine exact values)
Step 2: Find when f(x) = 1
Find where graph intersects y=1 line
Step 3: Domain and Range
Domain: All x-values shown on graph
Range: All y-values shown on graph
Step 4: Image of 6
Find f(6) value from graph
✅ Note: Exact answers depend on the graph provided
Let f(x) = 2x + 5. Find f(x) + f(-2).
Step 1: Compute f(-2)
f(-2) = 2(-2) + 5 = -4 + 5 = 1
Step 2: Add to f(x)
f(x) + f(-2) = (2x + 5) + 1 = 2x + 6
A function f is defined by f(x) = 2x - 3. Find:
(i) [f(0) + f(1)]/2
(ii) x such that f(x) = 0
(iii) x such that f(x) = x
(iv) x such that f(x) = -f(1)
(i) [f(0) + f(1)]/2
f(0) = -3, f(1) = -1 → (-3 + -1)/2 = -2
(ii) f(x) = 0
2x - 3 = 0 → x = 3/2
(iii) f(x) = x
2x - 3 = x → x = 3
(iv) f(x) = -f(1)
f(1) = -1 → -f(1) = 1
2x - 3 = 1 → x = 2
An open box is made from a 24cm square by cutting equal squares (x) from corners and folding.
2 cm
Step 1: Understand dimensions
When cutting squares of side x:
- Box height = x
- Box length = 24 - 2x
- Box width = 24 - 2x
Step 2: Volume formula
Volume = length × width × height = (24 - 2x) × (24 - 2x) × x
✅ Final Answer:
V(x) = x(24 - 2x)²
A function f is defined by f(x) = 3 - 2x. Find x such that f(2x) = f(f(x)).
Step 1: Compute f(2x)
f(2x) = 3 - 2(2x) = 3 - 4x
Step 2: Compute f(f(x))
f(x) = 3 - 2x
f(f(x)) = 3 - 2(3 - 2x) = 3 - 6 + 4x = -3 + 4x
Step 3: Set equal and solve
3 - 4x = -3 + 4x
6 = 8x → x = 3/4
A plane flies at 500 km/h. Express distance d as function of time t.
Step 1: Understand relationship
Distance = Speed × Time
✅ Final Answer:
d(t) = 500t
Data shows relationship between forehand length (x) and height (y) as y = ax + b.
(i) Is this relation a function?
(ii) Find a and b
(iii) Find height when x=40cm
(iv) Find forehand length when height=53.3 inches
(i) Is it a function?
Yes, each x maps to exactly one y
(ii) Find a and b
Using sample points (35,56) and (40,60):
56 = 35a + b
60 = 40a + b
Solving gives a = 0.8, b = 28
(iii) Height when x=40cm
y = 0.8(40) + 28 = 60 inches
(iv) Forehand when y=53.3 inches
53.3 = 0.8x + 28 → x ≈ 31.625 cm