Question

How do you graph, find any intercepts, domain and range of `f(x)=(1/4)^(x+3)-2`?

Answer 1

Doman is `(-oo,oo)`, range is `(-2,oo)`, `y`-intercept is `-1 63/64` and `x`-intercept is `-7/2`.

Explanation:

`f(x)=(1/4)^(x+3)-2`

`=(4^(-1))^(x+3)-2`

`=(2^2)^(-x-3)-2`

`=2^(-2x-6)-2`

As any power of `2` is possible, there are no limitations on value of `x` and doman is `(-oo,oo)`

However, as `x->oo`, as `2^(-oo)->0`, value of `f(x)` cannot go below `-2` and range is `(-2,oo)`

Further when `x=0`, `f(x)=2^(-6)-2=1/64-2=-1 63/64` and hence `y`-intercept is `-1 63/64` and as `f(x)=0` is at `2^(-2x-6)=2` or `-2x-6=1` i.e. `x=-7/2`, `x`-intercept is `-7/2`.

The graph appears as follows.

graph{(1/4)^(x+3)-2 [-10, 10, -5, 5]}