COMC Functions and Sequences 1 ā Questions and Answers
Question 1: A function f satisfies f(x + y) = f(x) + f(y) for all real numbers x and y, and f(1) = 4. What is f(7)?
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- 24
- 28 (Correct answer)
- 32
Correct answer: 28
By the Cauchy functional equation, f(n) = 4n for all integers, so f(7) = 4 Ć 7 = 28.
Question 2: If f(2x) = f(x) + x for all x > 0 and f(1) = 0, what is f(8)?
- 5
- 6
- 7 (Correct answer)
- 8
Correct answer: 7
f(2) = f(1)+1 = 1, f(4) = f(2)+2 = 3, f(8) = f(4)+4 = 7.
Question 3: Let f(x) = ax + b with a > 0. If f(f(x)) = 9x + 16, what is a + b?
- 5
- 6
- 7 (Correct answer)
- 8
Correct answer: 7
f(f(x)) = aĀ²x + b(a+1) = 9x + 16, so aĀ² = 9 ā a = 3, then 4b = 16 ā b = 4, giving a + b = 7.
Question 4: How many functions f: {1, 2, 3} ā {1, 2, 3} satisfy f(f(x)) = x for all x in the domain?
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- 4 (Correct answer)
- 5
- 6
Correct answer: 4
Such involutions are: the identity, swap(1,2) fix 3, swap(1,3) fix 2, swap(2,3) fix 1 ā exactly 4 functions.
Question 5: The function f satisfies f(1) = 1 and f(n+1) = f(n) + 2n for all positive integers n. What is f(6)?
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- 28
- 31 (Correct answer)
- 36
Correct answer: 31
f(2)=3, f(3)=7, f(4)=13, f(5)=21, f(6)=31; equivalently f(n) = nĀ² ā n + 1.
Question 6: Let g(x) = xĀ² ā 3 and f(x) = 2x + 1. For how many integer values of x does f(g(x)) = g(f(x))?
- 0 (Correct answer)
- 1
- 2
- 3
Correct answer: 0
f(g(x)) = 2xĀ²ā5 and g(f(x)) = 4xĀ²+4xā2; setting equal gives 2xĀ²+4x+3 = 0, which has discriminant 16ā24 < 0, so no real solutions.
Question 7: A function f defined for positive reals satisfies f(x/y) = f(x) ā f(y) for all x, y > 0, and f(4) = 2. What is f(64)?
- 4
- 5
- 6 (Correct answer)
- 8
Correct answer: 6
The rule implies f is logarithmic: f(4^k) = 2k, so f(64) = f(4Ā³) = 3 Ć 2 = 6.
COMC Functions and Sequences 1
A function f satisfies f(x + y) = f(x) + f(y) for all real numbers x and y, and f(1) = 4.
What is f(7)?