Question:
If \( f(x) = \frac{3^x}{3^x + \sqrt{3}} \), then \( f(x) + f(1-x) \) is equal to
If \( f(x) = \frac{3^x}{3^x + \sqrt{3}} \), then \( f(x) + f(1-x) \) is equal to
Show Hint
Check symmetry \( f(x) + f(1-x) \) — often simplifies to constant.
Updated On: Apr 21, 2026
- \( \sqrt{3} \)
- \( \frac{1}{\sqrt{3}} \)
- \( 2\sqrt{3} \)
- \( 1 \)
- \( 0 \)
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The Correct Option is D
Solution and Explanation
Concept:
Use substitution symmetry.
Step 1: Write \( f(1-x) \).
\[ f(1-x) = \frac{3^{1-x}}{3^{1-x} + \sqrt{3}} \] \[ = \frac{3 \cdot 3^{-x}}{3 \cdot 3^{-x} + \sqrt{3}} \]
Step 2: Simplify expression.
Let \( t = 3^x \), then: \[ f(x) = \frac{t}{t+\sqrt{3}}, \quad f(1-x)=\frac{3/t}{3/t + \sqrt{3}} \] \[ = \frac{3}{3 + \sqrt{3}t} \]
Step 3: Add.
\[ f(x) + f(1-x) = 1 \]
Step 1: Write \( f(1-x) \).
\[ f(1-x) = \frac{3^{1-x}}{3^{1-x} + \sqrt{3}} \] \[ = \frac{3 \cdot 3^{-x}}{3 \cdot 3^{-x} + \sqrt{3}} \]
Step 2: Simplify expression.
Let \( t = 3^x \), then: \[ f(x) = \frac{t}{t+\sqrt{3}}, \quad f(1-x)=\frac{3/t}{3/t + \sqrt{3}} \] \[ = \frac{3}{3 + \sqrt{3}t} \]
Step 3: Add.
\[ f(x) + f(1-x) = 1 \]
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