Question

How do you find the asymptotes for `4^(x-5)-5`?

Answer 1

See below.

Explanation:

Vertical asymptotes occur where the function is undefined. This particular function is defined for all x, so no vertical asymptotes. To find any horizontal asymptotes will need to examine the end behaviour as

`x-> +-oo`

as `x->oo` , `color(white)(88)4^(x-5)-5->oo`

For `x<5`:

`4^(x-5)-5` becomes `1/4^(x-5)-5`

as `x->oo` ,`color(white)(88)1/4^(x-5)-5->-5`

( we used positive infinity here, since we rewrote the expression )

So the horizontal asymptote is the line:

`y= -5`

graph{4^(x-5)-5 [-16.02, 16.01, -8.01, 8.01]}