# How do you graph f(x)=2^x+1 and state the domain and range?

##### 1 Answer
Aug 30, 2017

Domain: $\left(- \infty , + \infty\right)$ Range: $\left(1 , + \infty\right)$
Graph: See below

#### Explanation:

$f \left(x\right) = {2}^{x} + 1$

The graph of $f \left(x\right)$ is the standard graph of ${2}^{x}$ transformed ("shifted") one unit positive ("up") on the $y -$axis.

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

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

Consider:

(i) ${\lim}_{x \to - \infty} f \left(x\right) = 0 + 1 = 1$

(ii) $f \left(x\right)$ has no finite upper bound

Hence, the range of $f \left(x\right)$ is $\left(1 , + \infty\right)$

The domain and range of $f \left(x\right)$ may be inferred from its graph below.

graph{2^x+1 [-12.34, 7.655, -1.88, 8.12]}