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Inverse and composition of functions

  • Thread starter terryds
  • Start date
  • #1
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Homework Statement



If ##f(2x-1)= 6x + 15## and ##g(3x+1)=\frac{2x-1}{3x-5}##, then what is ##f^{-1}\circ g^{-1}(3)## ?

a) -2
b) -3
c) -4
d) -5
e) -6

The Attempt at a Solution



I think the f inverse and g inverse is
##f^{-1}(6x+15)= 2x-1##
##g^{-1}(\frac{2x-1}{3x-5})=3x+1##

and,##f^{-1}\circ g^{-1}(3)=f^{-1}(g^{-1}(3))##

Then,

I equate ##\frac{2x-1}{3x-5}=3##
So, I get x = 2

So, ##g^{-1}(3)=3(2)+1=7##

Then, I equate ##6x+15=7##
And, I get x = 8/6 or 4/3

So, ##f^{-1}(7)=2(\frac{4}{3})-1=\frac{5}{3}## which does not appear in the options..
Please help me figure out what's wrong
 

Answers and Replies

  • #2
694
114
Let's deal with ##g## first. Let the input (##3x+1##) be equal to ##u##. Then ##x=\frac{u-1}{3}##. So you have g(u) = something, and you have to find that something by expressing the output in terms of ##u##. Then try inverting the function.
 
  • #3
392
13
Let's deal with ##g## first. Let the input (##3x+1##) be equal to ##u##. Then ##x=\frac{u-1}{3}##. So you have g(u) = something, and you have to find that something by expressing the output in terms of ##u##. Then try inverting the function.
I get ##g(u)=\frac{2u-5}{3u-18}##
Then ##g^{-1}(x)=\frac{18x-5}{3x-2}##
So, ##g^{-1}(3)=7##

Then it'll be ##f^{-1}(7)=\frac{5}{3}##
The answer is -5 according to the book.. But, it doesn't give me the explanation.. :frown:
 
  • #4
Samy_A
Science Advisor
Homework Helper
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Then, I equate ##6x+15=7##
And, I get x = 8/6 or 4/3
That should be ##x=-4/3##. Still doesn't give the answer from the book though.
 
  • #5
392
13
That should be ##x=-4/3##. Still doesn't give the answer from the book though.
Yup, perhaps the question is wrong
 
  • #6
Samy_A
Science Advisor
Homework Helper
1,241
510
Yup, perhaps the question is wrong
Or I'm missing something. I redid your calculation, and also did it like @PWiz suggested (for the two functions), but with both methods I get ##-11/3## as result.
 
  • #7
694
114
I just did the calculation myself. I'm getting -11/3 as the answer. I'm guessing something is wrong with the answer given.
EDIT: I see that @Samy_A has gotten the same result. That pretty much confirms that something is wrong with the question, since we both can't get the same wrong answer.
 

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