Find domain, range, x and y intercept, Horizontal and vertical asymptote? How do I do it?

f(x)=3^x-9

Question

Find domain, range, x and y intercept, Horizontal and vertical asymptote? How do I do it?

f(x)=3^x-9

anonymous · Answer

The domain is all the possible values you can have for x.  There is no value for x where we would have "problems" like dividing by 0 or having a negative under the square root.  Therefore, the domain of that function is all real numbers or (-infinity, infinity).  The range is all the possible values you can have for y.  If you solve for x in your equation, you'll have:

$$f(x)+9=3^x$$
$$x = \log_{3}[f(x)+9] $$

Now, what possible values can we have for f(x)?  Since a log cannot have 0 or negative numbers on the inside, f(x) > -9.  So, the range is (-9, infinity)

There are no vertical asymptotes for exponential functions (just kind of a given if you are precalculus).  The horizontal asymptote occurs on the y = -9 when we look at range.

anonymous · Answer

What about the x and y int? On my solution set i have x-int: (2,0) and y-int: (0,-8). How do you get those answers?

I understand the other parts.

anonymous · Answer

Oh whoops.  Forgot about the intercepts.  To find the x-intercept, let y = 0 to get 0 = 3^x - 9; solve for x to get x = 2, or the point (2, 0).  To find the y-intercept, let x = 0 to get f(x) = 3^0 - 9 = 1 - 9 = -8.  So, the y-intercept is at the point (0, -8)