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Prove that f (x) = 2x^3 + 5 and g (x) = sqrt3((x-5)/2) are inverses of one another

Apr 20, 2022

#1
+1

To find the inverse of  f(x)  =  2x3 + 5  follow these steps:

1)  Replace  'f(x)'  with  'y':  y  =  2x3 + 5

2)  Interchance  'x'  and  'y':  x  =  2y3 + 5

3)  Solve this equation for y:  x  =  2y3 + 5

x - 5  =  2y3

(x - 5) / 2  =  y3

[ (x - 5) / 2 ]1/3  =  y

4)  Replace  'y'  with  'f-1(x)'  indicating that this equation is the inverse:  f-1(x)  =   [ (x - 5) / 2 ]1/3

This shows that  g(x)  is the inverse of  f(x).

If you want to, you can follow these steps to find the inverse of  g(x), showing that  f(x)  is the inverse of  g(x).

Apr 20, 2022
#2
+1

Thanks geno .....here's another way

If    f (g)  = x      and g (f)  = x   then they are inverses

f (g)  =     2 [ [ (x - 5) / 2 ] ^1/3 ]^3    +  5    =      2  ( x  -5) / 2  + 5  =   x - 5 + 5  = x

g (f)  =  (  [ 2x^3 + 5 - 5 ] / 2 )^(1/3)  =   [2x^3/ 2] ^(1/3)   [ x^3]^(1/3)  = x

So....inverses    Apr 21, 2022