Discussion
If \(f(x) = \dfrac{3^{x}}{3^{x} + \sqrt{3}}\) , then \(f(x) + f(1 - x)\) is equal to
Answer: D
\(f(x) + f(1 - x) = \) 1.
Simplify \(f(1 - x)\) by multiplying top and bottom by \(3^{x}\):
\[ \begin{aligned} f(1 - x) &= \frac{3 \cdot 3^{-x}}{3 \cdot 3^{-x} + \sqrt{3}} = \frac{3}{3 + \sqrt{3}\,3^{x}} \\ &= \frac{\sqrt{3}}{\sqrt{3} + 3^{x}} \end{aligned} \]
Adding \(f(x) = \frac{3^{x}}{3^{x} + \sqrt{3}}\), the numerators sum to the common denominator, so the total is 1. Check at \(x = \frac{1}{2}\): \(f(\frac{1}{2}) = \frac{1}{2}\), and twice that is 1. The surd options mistake the constant \(\sqrt{3}\) for the answer.
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