g(x)=log(base3)(2-x) how can I evaluate and graph this equation to find the domain and range?

Question

I asked about this question in the homework section it got 2 responses quickly but I forgot to state an important part of the problem in my original question and there doesnt seem to be a way to reply to your question so I thought I would ask it here:
I have a question on my homework that asks to state the domain and range of the function g(x)=log(base3)(2-x)
Using the change of bases property I rewrote it as
f(x)=log(2-x)/log(3) and entered that function into my graphing calculator. The graph I got back was a downward sloping graph that ended at x=2 making the domain (-infinity, 2). Unfortunately this question is mulitple choice and the only 2 possible domains are (-infinity, infinity) and (0, infinity). I have been working on this problem for hours looking through my book for any examples with a base that isnt 10 or e and addition or subtraction of a variable and a constant but there isnt anything. Anyone see anything wrong with how I rewrote the equation?
Thanks

Answer 1

Maybe you've read it wrong, somehow? The change of base formula is valid for any positive base except for 1, so it will work for base 3.

Algebraically, log(2-x) is defined (for any base) for 2-x>0, which works out to x<2, namely the domain should be (-infinity, 2). Even if there were absolute values in the problem (that is, log|2-x| instead of log(2-x) ) then it would be undefined for x=2 so neither of the answers you gave would be correct.

The range will be (-infinity, infinity), however so maybe you are reading the answers wrong somehow? Hmm.

Answer 2

As negative numbers don´t have log( in any base)2-x>0 and x<2 is the domain
As the base is >1 the range is from -infinity ( as x=>2-) to + infinity (as x=> to -infinity)

Answer 3

it is merely an incomplete piecewise function. The area is what's written down. D: x < -3 the style is each value under f(-3) by fact the slope is helpful. R: y < -a million The graph is merely the time-honored line y = x + 2 with an open dot at (-3, -a million). it is the shrink to what x can circulate as much as. do no longer draw something of line it relatively is to the suited, and subsequently above, the line you have already drawn.