Find Function f(x)

🎥 Video Explanation

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📝 Question

If \( f : A \to B \) is defined by

\[ 3^{f(x)} + 2^{-x} = 4 \]

and \(f\) is bijective, find \(f(x)\).

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✅ Solution

🔹 Step 1: Isolate \(f(x)\)

\[ 3^{f(x)} = 4 – 2^{-x} \] —

🔹 Step 2: Take log base 3

\[ f(x) = \log_3\left(4 – 2^{-x}\right) \] —

🔹 Step 3: Domain Condition

For logarithm to exist:

\[ 4 – 2^{-x} > 0 \]

\[ 2^{-x} < 4 \]

\[ 2^{-x} < 2^2 \Rightarrow -x < 2 \Rightarrow x > -2 \]

So domain: \[ A = (-2, \infty) \]

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🔹 Step 4: Range

As \(x \to \infty\), \(2^{-x} \to 0\): \[ f(x) \to \log_3(4) \]

As \(x \to -2^+\), \(2^{-x} \to 4\): \[ f(x) \to \log_3(0) = -\infty \]

So range: \[ B = (-\infty, \log_3 4) \]

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🔹 Final Answer

\[ \boxed{f(x)=\log_3(4 – 2^{-x})} \]

Domain: \((-2, \infty)\)
Range: \((-\infty, \log_3 4)\)