Question 80375
Given:
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{{{f(x) = (2/3)x - (3/5)}}}
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Note that I presume you meant {{{(2/3)*x}}} and not {{{2/3x}}} which is what you wrote
if the rules of algebraic convention are followed.
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There are four steps to finding an inverse of the function. These steps are:
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1. Replace f(x) with y
2. Then replace y by x and also change x to y
3. Solve for y
4. Finally replace y by {{{f^-1(x)}}}
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Let's do it by following these 4 steps. Start with the given:
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{{{f(x) = (2/3)x - (3/5)}}}
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Replace f(x) by y to get:
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{{{y = (2/3)x - (3/5)}}}
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Step 1 is done.  Next do step 2 by changing y to x and also changing x to y.  When you
do that the equation from step 1 becomes:
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{{{x = (2/3)y - (3/5)}}}
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Step 2 is now done.  On to step 3 which tells you to solve for this equation for y. 
You can isolate the term that contains y. Do that by adding {{{3/5)}}} to both sides of 
the equation to get:
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{{{x + (3/5) = (2/3)y -(3/5) + (3/5)}}}
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Notice that the terms {{{-3/5}}} and {{{3/5}}} cancel each other on the right side so 
the equation is reduced to:
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{{{x + (3/5) = (2/3)y}}}
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You can get rid of the denominator 5 on the right side by multiplying all the terms on
both sides of this equation by 5.  Multiplying everything by 5 results in:
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{{{5x + 3 = (10/3)*y }}}
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Now get rid of the denominator of 3 on the right side by multiplying everything on both
sides by 3 to get:
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{{{15x + 9 = 10y}}}
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Now solve for y by dividing both sides by 10 and the result is:
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{{{(15x + 9)/10 = y}}}
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Step 3 is done.  Now replace y with {{{f^-1(x)}}} and the answer becomes:
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{{{f^-1(x)= (15x + 9)/10}}}
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Presuming that I interpreted your problem correctly, this is the answer you were asked to
find.  I'm not sure where you got the answer that you did. Even if I interpreted it incorrectly,
the four step process is correct, and you can use this same method to solve any problem with
inverting a function.
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