Functions Question - None

Question

If

f_1(x) = 2^{f_2(x)}

, where

f_2(x) = 2012^{f_3(x)}

, where

f_3(x) = \left( \frac{1}{2013} \right)^{f^{(x)}}

,

f_4(x) = \log_{2013}\log_{x}2012

, then the range of

f_1(x)

is -

Accepted Answer

Step 1: Analyze f₄(x) = log₂₀₁₃(log_x(2012))

For f₄(x) to be defined: log_x(2012) > 0, which requires x > 1 (since log_x(2012) = ln(2012)/ln(x), we need ln(x) > 0).

When x > 1: log_x(2012) can range from 0⁺ to ∞ as x varies from ∞ to 1⁺.

Step 2: Find range of f₄(x)

As log_x(2012) ranges over (0, ∞), log₂₀₁₃(log_x(2012)) ranges over (-∞, ∞).

Step 3: Analyze f₃(x) = (1/2013)^(f₄(x))

Since (1/2013)^t is an exponential function with base 0 < 1/2013 < 1:

• When f₄(x) → ∞, f₃(x) → 0⁺

• When f₄(x) → -∞, f₃(x) → ∞

• When f₄(x) = 0, f₃(x) = 1

Thus, f₃(x) ∈ (0, ∞) for all valid x.

Step 4: Analyze f₂(x) = 2012^(f₃(x))

Since f₃(x) ∈ (0, ∞) and 2012 > 1:

• When f₃(x) → 0⁺, f₂(x) → 2012⁰ = 1

• When f₃(x) → ∞, f₂(x) → ∞

Thus, f₂(x) ∈ (1, ∞).

Step 5: Analyze f₁(x) = 2^(f₂(x))

Since f₂(x) ∈ (1, ∞) and 2 > 1:

• When f₂(x) → 1⁺, f₁(x) → 2¹ = 2

• When f₂(x) → ∞, f₁(x) → ∞

Thus, f₁(x) ∈ (2, ∞).

∴ Answer: A

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