Notice that,
[tex]f(x)=3^{x-1}-6=3^x\cdot3^{-1}-6=\frac{3^x}{3}-6[/tex]
And there are no restrictions for the values that x can take. The domain is the whole set of real numbers.
Now, we need to check for the limits when x->+/- infinite, as follows:
[tex]\begin{gathered} \lim _{x\to\infty}3^x=\infty \\ \lim _{x\to-\infty}3^x=\lim _{x\to\infty}\frac{1}{3^x}=0 \end{gathered}[/tex]
Then, the range of 3^x is (0, infinite).
Finally, we can get the range of function f(x):
[tex]\lim _{x\to\infty}f(x)=\frac{1}{3}(\lim _{x\to\infty}3^x)-6=\frac{1}{3}\infty-6=\infty[/tex][tex]\lim _{x\to-\infty}f(x)=\frac{1}{3}(\lim _{x\to-\infty}3^x)-6=\frac{1}{3}\cdot0-6=-6[/tex]
Then,
[tex]\begin{gathered} The\text{ range of }f(x)\text{ is} \\ Range=(-6,\infty) \end{gathered}[/tex]
