SOLUTION: Given the function f(x)=3x^3+2, find the value of x so that f^-1(x)=4. Thank you.

Algebra ->  Rational-functions -> SOLUTION: Given the function f(x)=3x^3+2, find the value of x so that f^-1(x)=4. Thank you.      Log On


   

Question 919810: Given the function f(x)=3x^3+2, find the value of x so that f^-1(x)=4. Thank you.
Found 2 solutions by MathLover1, richard1234:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=3x%5E3%2B2,
first find f%5E-1%28x%29
f%28x%29=3x%5E3%2B2 .....since f%28x%29=y we have
y=3x%5E3%2B2...swap x and y
x=3y%5E3%2B2 .........solve for y
x-2=3y%5E3
%281%2F3%29x-2%2F3=y%5E3
y=root%283%2C%281%2F3%29x-2%2F3%29 => f%5E-1%28x%29
f%5E-1%28x%29=root%283%2C%281%2F3%29x-2%2F3%29

now find the value of x so that f%5E-1%28x%29=4:

4=root%283%2C%281%2F3%29x-2%2F3%29 ...both sides raise to power of 3

4%5E3=%28root%283%2C%281%2F3%29x-2%2F3%29%29%5E3

64=%281%2F3%29x-2%2F3 ...solve for x
64%2B2%2F3=%281%2F3%29x
192%2F3%2B2%2F3=%281%2F3%29x
194%2F3=x%2F3 ...if denominators same, then nominators are same too
highlight%28x=194%29

check if f%5E-1%28x%29=4

f%5E-1%28194%29=root%283%2C%281%2F3%29194-2%2F3%29

f%5E-1%28194%29=root%283%2C194%2F3-2%2F3%29
f%5E-1%28194%29=root%283%2C192%2F3%29
f%5E-1%28194%29=root%283%2C64%29
f%5E-1%28194%29=root%283%2C4%5E3%29
f%5E-1%28194%29=4



Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
f(4) = 3*4^3 + 2 = 194

Taking f^(-1) of both sides, 4 = f^(-1)(194), so x in this case is 194. Note that f is bijective (more precisely one-to-one), so it has an inverse function.