# How do you determine if the equation f(x) = 2(1/2)^x represents exponential growth or decay?

Sep 16, 2016

$f \left(x\right)$ is decay

#### Explanation:

$\textcolor{b r o w n}{\text{Moving to the right on the number line}}$

As $x$ becomes ever greater then ${\left(\frac{1}{2}\right)}^{x}$ becomes less and less but still positive.

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$\textcolor{b r o w n}{\text{Moving left on the number line}}$

When $x < 0$ we have $2 {\left(\frac{2}{1}\right)}^{| x |}$

The more and more negative $x$ becomes the greater $2 {\left(\frac{2}{1}\right)}^{| x |}$
becomes. However in doing so we are moving further from 0 in the negative direction. Consequently changing this direction so that you move closer to 0 then $2 {\left(\frac{2}{1}\right)}^{| x |}$ is decreasing
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Thus $2 {\left(\frac{1}{2}\right)}^{x}$ is always decreasing as you move to the right.

${\lim}_{x \to \infty} 2 {\left(\frac{1}{2}\right)}^{x} \to 0$