Show \(f(x)=\dfrac{1}{2}-\dfrac{1}{2^x+1}\) is an odd function. (3 marks) --- 10 WORK AREA LINES (style=lined) --- \(\text{Odd}\ \ \Rightarrow \ \ f(-x)=-f(x)\) \(\begin{aligned} \(\text{Odd}\ \ \Rightarrow \ \ f(-x)=-f(x)\) \(\begin{aligned}
f(x) & =\dfrac{1}{2}-\dfrac{1}{2^x+1} \\
f(-x) & =\dfrac{1}{2}-\dfrac{1}{2^{-x}+1} \times \dfrac{2^x}{2^x} \\
& =\dfrac{1}{2}-\dfrac{2^x}{1+2^x} \\
& =\dfrac{1}{2}-\dfrac{2^x+1-1}{2^x+1} \\
& =\dfrac{1}{2}-1+\dfrac{1}{2^x+1} \\
& =-\dfrac{1}{2}+\dfrac{1}{2^x+1} \\
& =-f(x)
\end{aligned}\)
\(\therefore f(x) \text { is odd.}\)
f(x) & =\dfrac{1}{2}-\dfrac{1}{2^x+1} \\
f(-x) & =\dfrac{1}{2}-\dfrac{1}{2^{-x}+1} \times \dfrac{2^x}{2^x} \\
& =\dfrac{1}{2}-\dfrac{2^x}{1+2^x} \\
& =\dfrac{1}{2}-\dfrac{2^x+1-1}{2^x+1} \\
& =\dfrac{1}{2}-1+\dfrac{1}{2^x+1} \\
& =-\dfrac{1}{2}+\dfrac{1}{2^x+1} \\
& =-f(x)
\end{aligned}\)
\(\therefore f(x) \text { is odd.}\)
L&E, 2ADV E1 EQ-Bank 2
Show \(f(x)=\dfrac{1}{2}-\dfrac{1}{2^x+1}\) is an odd function. (3 marks) --- 10 WORK AREA LINES (style=lined) ---