# How do you find the domain and range of f(x) = 3(1/2)^x?

Dec 12, 2017

Domain: $x \in \mathbb{R}$
Range: $f \left(x\right) > 0$

#### Explanation:

We can first sketch the function to enable us to determine the domain and range:

graph{3*(1/2)^x [-9.16, 10.84, -2.12, 7.88]}

I am assuming you are aware of how to sketch this function

Now we can consider the domain:

The fucntion is defined for all values of $x$ ie, it will always output $f \left(x\right)$ where $f \left(x\right)$ is real, or we can write as $\forall x , f \left(x\right) \in \mathbb{R}$

$\implies x \in \mathbb{R}$

Now we can find the range:

We see that there is a asymptote at $y = 0$ we the function $f \left(x\right)$ gets closer to $0$ as $x \to \infty$ but will never touch $y = 0$

But then $f \left(x\right)$ can take on all the other positive real numbers:

$\implies f \left(x\right) > 0$