# How do you graph y=(0.5)^x and state the domain and range?

Dec 23, 2016

graph{(0.5)^x [-10, 10, -5, 5]}

domain: $\left(- \infty , \infty\right)$
range: $\left(0 , \infty\right)$

#### Explanation:

$y = {\left(0.5\right)}^{x}$

domain (the values that $x$ could take):

$x$ could be substituted for all real numbers, including $\pi$ and $e$.

this means that the values that $x$ could take go from $- \infty$ to $\infty$.

$\therefore - \infty$ and $\infty$ are the domain, and are written as $\left(- \infty , \infty\right)$

range (the values that $y$ could take):

$y = {\left(0.5\right)}^{x}$

the roots (square, cube, etc.) of negative numbers are lateral ($i$ or a factor/multiple of this), and 0.5 is a real number-

$\therefore {\left(0.5\right)}^{x} > 0$.

since the value of ${\left(0.5\right)}^{x}$ decreases as $x$ increases, $y$ gradually goes towards $0$.

although it does not actually reach $0$, $0$ is its limit as $x$ approaches $\infty$.

this means that the values $y$ could take go from $0$ to $\infty$.

$\therefore 0$ and $\infty$ are the range, and are written as $\left(0 , \infty\right)$.