SOLUTION: Having some trouble with this problem can someone give me a hand? I'm really just having trouble getting the x and y intercepts my teacher confused me when he explained it. So if s

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Question 865322: Having some trouble with this problem can someone give me a hand? I'm really just having trouble getting the x and y intercepts my teacher confused me when he explained it. So if someone could explain how to get them it would really help.
let f(x)=log[base 3](9-3x)
Find
Domain: Which I think is (-oo,3)
Range: All real numbers
Vertical Asymptote:I think its x=3
X-Intercept:?
Y-Intercept:?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You have the correct domain and range. Also, you have the correct vertical asymptote. Nice work on getting those right.


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To find the x-intercept, you plug in y = 0 or f(x) = 0 and solve for x.


f%28x%29+=+log%283%2C%289-3x%29%29


0+=+log%283%2C%289-3x%29%29 Plug in f(x) = 0. Remember that f(x) and y are interchangeable.


3%5E0+=+9-3x


1+=+9-3x


1-9+=+-3x


-8+=+-3x


-8%2F%28-3%29=x


8%2F3=x


x=8%2F3


The x-intercept is 8%2F3 which is the point (8/3, 0)


Note: since 8%2F3=2.6666667 approximately, this means that the x-intercept is approximately the point (2.6666667,0)


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To find the y-intercept, you plug in x = 0 and solve for y (or f(x)).


f%28x%29+=+log%283%2C%289-3x%29%29


f%280%29+=+log%283%2C%289-3%280%29%29%29 Plug in x = 0.


f%280%29+=+log%283%2C%289-0%29%29


f%280%29+=+log%283%2C%289%29%29


f%280%29+=+log%283%2C%283%5E2%29%29


f%280%29+=+2%2Alog%283%2C%283%29%29


f%280%29+=+2%2A1


f%280%29+=+2


The y-intercept is 2 which is the point (0,2)


Here is the graph of f%28x%29+=+log%283%2C%289-3x%29%29 (the red curve). The blue vertical dashed line is the vertical asymptote. The red curve will never touch or cross the blue dashed line (even though it looks like it does). It will just get closer and closer to it.