Just look at it

Algebra Level 4

If $g\left( x \right) =x-1+\dfrac { 2 }{ 1+{ e }^{ x } }$ , then $g$ is a/an:

Odd function Even function Neither odd nor even function

2 solutions

Chew-Seong Cheong
Jan 23, 2016

$\begin{aligned} g(x) & = x - 1 + \frac{2}{1+e^x} \\ \Rightarrow g(-x) & = -x - 1 + \frac{2}{1+e^{-x}} \\ & = -x - 1 + \frac{2e^x}{1+e^x} \\ & = - \left(x - 1 + \frac{2}{1+e^x} \right) - 1 + \frac{2}{1+e^x} - 1 + \frac{2e^x}{1+e^x} \\ & = - \left(x - 1 + \frac{2}{1+e^x} \right) - 2 + \frac{2(1+e^x}{1+e^x} \\ & = -g(x) + 0 \end{aligned}$

$\Rightarrow g(-x) = - g(x) \quad \Rightarrow g(x) \text{ is } \boxed{\text{odd}}.$

Mohtasim Nakib
Jan 24, 2016

Just look at it

2 solutions

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