Question

Will f(x)=a^x-4 ever cross the x axis

Answer 1

That means will we every get y=0
or f(x)=0

solve
0=a^x-4
not sure if
0=(a^x)-4 or
0=a^(x-4)

if it is the first one then
when a^x=4, then it will work (it will cross)

if it is the 2nd then it will never cross, just come very close (when x=- infinity, we get 1/(a^infinity) which approaches 0 but doesn't cross




if it is f(x)=aˣ-4 then it will cross
if it is f(x)=aˣ⁻⁴ then it will not cross

Answer 2

Let's consider the principal function first:
$f(x) = a^{x}$

For an exponential, we know that there is a horizontal asymptote at f(x) = 0. Thus, we know that $f(x) = a^{x} > 0, x \in \mathbb{R}$

By moving the graph down 4 units, we have changed the range of the graph. For $f(x) = a^{x}$, there is a horizontal asymptote at y = 0, then there is a horizontal asymptote at y  = 4 for $f(x) = a^{x} - 4$

Thus, we can conclude for any value of a, except for a = 1, we know that it will cross the x-axis.