# How do you graph f(x)=(3/2)^x-2 and state the domain and range?

Dec 19, 2017

Domain: $\left(- \infty , + \infty\right)$
Range: $\left(- 2 , + \infty\right)$

#### Explanation:

$f \left(x\right) = {\left(\frac{3}{2}\right)}^{x} - 2$

$f \left(x\right)$ is the exponential increasing graph of $y = {\left(\frac{3}{2}\right)}^{x}$ transformed ("shifted") by 2 units negative ("down") on the $y -$axis.

$f \left(x\right)$ is defined $\forall x \in \mathbb{R}$

Hence, the domain of $f \left(x\right)$ is $\left(- \infty , + \infty\right)$

Consider, ${\lim}_{x \to - \infty} f \left(x\right) = - 2$

also, $f \left(x\right)$ has no finite upper bound.

Hence, the range of $f \left(x\right)$ is $\left(- 2 , + \infty\right)$

We can deduce these results from the graph of $f \left(x\right)$ below.

graph{(3/2)^x -2 [-10, 10, -5, 5]}