SOLUTION: G(t)= (T^3+8)/(T^3-1) 1. Find G^-1(t) 2. What is the domain and range of G. in each case, show or explain how you determine you answer. Thanks in advance

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: G(t)= (T^3+8)/(T^3-1) 1. Find G^-1(t) 2. What is the domain and range of G. in each case, show or explain how you determine you answer. Thanks in advance      Log On


   

Question 356329: G(t)= (T^3+8)/(T^3-1)
1. Find G^-1(t)
2. What is the domain and range of G. in each case, show or explain how you determine you answer.
Thanks in advance
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
G%28T%29=%28T%5E3%2B8%29%2F%28T%5E3-1%29+
To find the inverse, interchange the positions of G and T and solve for the new G.
The new G is the inverse.
highlight%28T%29=+%28%28highlight%28G%29%29%5E3%2B8%29%2F%28%28highlight%28G%29%29%5E3-1%29
T%28G%5E3-1%29=G%5E3%2B8
TG%5E3-T=G%5E3%2B8
TG%5E3-G%5E3=T%2B8
G%5E3%28T-1%29=T%2B8
G%5E3=%28T%2B8%29%2F%28T-1%29
highlight%28G%5Binv%5D%28T%29=root%283%2C%28T%2B8%29%2F%28T-1%29%29%29
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.
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The domain is all values of T where G%28T%29 is defined. The only problem is when the denominator equals zero since division by zero is undefined.
T%5E3-1=0
T=1
So the domain is all values of T except T=1 or in interval notation
Domain:(-infinity,1) U (1,infinity)
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graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C%28x%5E3%2B8%29%2F%28x%5E3-1%29%29
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The range is all values that G%28T%29 can have.
As you see from the graph, G%28T%29 can take on pretty much any value except G%28T%29=1.
Algrebraically you can see why it can't ever exactly equal 1.
%28T%5E3%2B8%29%2F%28T%5E3-1%29=1+
T%5E3%2B8=T%5E3-1+
8=-1
So the range can take on all values except G%28T%29=1 or in interval notation.
Range:(-infinity,1) U (1,infinity)