🧙♂️ The Magic of f ∘ g
When two functions f(x) and g(x) combine, they create a magical composition:
🌟 f ∘ g means f(g(x)) → First apply g, then apply f!
🌈 g ∘ f means g(f(x)) → First apply f, then apply g!
Problem 1: Basic Composition
For each pair of functions, find f ∘ g and g ∘ f. Check if they're equal!
- f(x) = x - 6, g(x) = x²
- f(x) = x/2, g(x) = 2x² - 1
- f(x) = 3/(x + 6), g(x) = 3 - x
- f(x) = 3 + x, g(x) = x - 4
- f(x) = 4x² - 1, g(x) = 1 + x
i)
f ∘ g = x² - 6,
g ∘ f = (x - 6)² →
Not equal!
ii)
f ∘ g = x² - 0.5,
g ∘ f = x²/2 - 1 →
Not equal!
iii)
f ∘ g = 3/(9 - x),
g ∘ f = 3 - 3/(x + 6) →
Not equal!
iv)
f ∘ g = x - 1,
g ∘ f = x - 1 →
Equal! ✨
v)
f ∘ g = 4(1 + x)² - 1,
g ∘ f = 1 + 4x² - 1 = 4x² →
Not equal!
Problem 2: Find the Magic Number k
Find k such that f ∘ g = g ∘ f:
- f(x) = 3x + 2, g(x) = 6x - k
- f(x) = 2x - k, g(x) = 4x + 5
i)
Set 18x - 3k + 2 = 18x + 12 - k →
-2k = 10 →
k = -5
ii)
Set 8x - 4k + 5 = 8x + 10 - 2k →
-2k = 5 →
k = -2.5
Problem 3: Inverse Magic
Show that for f(x) = 2x - 1 and g(x) = (x + 1)/2:
f ∘ g = g ∘ f = x
f ∘ g = f((x+1)/2) = 2((x+1)/2) - 1 = x
g ∘ f = g(2x - 1) = ((2x - 1) + 1)/2 = x
They're inverses! ✨
Problem 4: Find the Hidden Number
Given f(x) = x² - 1 and g(x) = x - 2,
Find a if g ∘ f(a) = 1
g ∘ f(a) = g(a² - 1) = (a² - 1) - 2 = a² - 3 = 1
a² = 4 → a = ±2
Problem 5: Range Explorer
For f(x) = 2x + 1 and g(x) = x², find:
The range of f ∘ g and g ∘ f
f ∘ g = f(x²) = 2x² + 1 → Range: [1, ∞)
g ∘ f = g(2x + 1) = (2x + 1)² → Range: [0, ∞)
Problem 6: Triple Composition
For f(x) = x² - 1, find:
i) f ∘ f ii) f ∘ f ∘ f
i) f ∘ f = f(x² - 1) = (x² - 1)² - 1 = x⁴ - 2x²
ii) f ∘ f ∘ f = f(x⁴ - 2x²) = (x⁴ - 2x²)² - 1 = x⁸ - 4x⁶ + 4x⁴ - 1
Problem 7: One-One Investigation
For f(x) = x⁵ and g(x) = x⁴:
Check if f, g are one-one and if f ∘ g is one-one
f(x) = x⁵ is one-one (strictly increasing)
g(x) = x⁴ is not one-one (g(1) = g(-1) = 1)
f ∘ g = f(x⁴) = (x⁴)⁵ = x²⁰ is not one-one (same output for x and -x)
Problem 8: Associative Property
Show that (f ∘ g) ∘ h = f ∘ (g ∘ h) for:
- f(x) = x - 1, g(x) = 3x + 1, h(x) = x²
- f(x) = x², g(x) = 2x, h(x) = x + 4
- f(x) = x - 4, g(x) = x², h(x) = 3x - 5
i) Both sides equal 3x² + 1 - 1 = 3x²
ii) Both sides equal (2(x + 4))² = 4(x + 4)²
iii) Both sides equal (3x - 5)² - 4
Problem 9: Function Reconstruction
Given f = {(-1,3), (0,-1), (2,-9)} is linear, find f(x)
Assume f(x) = ax + b
From (0,-1): b = -1
From (-1,3): -a - 1 = 3 → a = -4
Verify with (2,-9): -4(2) - 1 = -9 ✓
Solution: f(x) = -4x - 1
Problem 10: Real-World Application
In electrical circuits, a circuit C(t) is linear if it satisfies:
C(a·t₁ + b·t₂) = a·C(t₁) + b·C(t₂)
Show that C(t) = 3t is linear
Compute left side: C(a·t₁ + b·t₂) = 3(a·t₁ + b·t₂) = 3a·t₁ + 3b·t₂
Compute right side: a·C(t₁) + b·C(t₂) = a·3t₁ + b·3t₂ = 3a·t₁ + 3b·t₂
Both sides equal → C(t) is linear!