Question

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

Answer 1

Domain: `(-oo, +oo)` Range: `(1, +oo)`
Graph: See below

Explanation:

`f(x) = 2^x+1`

The graph of `f(x)` is the standard graph of `2^x` transformed ("shifted") one unit positive ("up") on the `y-`axis.

`f(x)` is defined `forall x in RR`

`:.` the domain of `f(x)` is `(-oo, +oo)`

Consider:

(i) `lim_(x->-oo) f(x) = 0+1 =1`

(ii) `f(x)` has no finite upper bound

Hence, the range of `f(x)` is `(1, +oo)`

The domain and range of `f(x)` may be inferred from its graph below.

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