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

Let's consider the principal function first:


For an exponential, we know that there is a horizontal asymptote at f(x) = 0. Thus, we know that

By moving the graph down 4 units, we have changed the range of the graph. For , there is a horizontal asymptote at y = 0, then there is a horizontal asymptote at y  = 4 for

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