SOLUTION: Given the function f(x)=-2c+cx-x^2 and f^-1(3)=-1, find c

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Question 1162043: Given the function f(x)=-2c+cx-x^2 and f^-1(3)=-1, find c
Found 3 solutions by MathLover1, ikleyn, Edwin McCravy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Given the function f%28x%29=-2c%2Bcx-x%5E2 and f%5E-1%283%29=-1, find c
first find inverse
f%28x%29=y
y=-2c%2Bcx-x%5E2 ...swap variables
x=-2c%2Bcy-y%5E2..........complete square
x=-2c%2B%28b%5E2%2Bcy-y%5E2%29-b%5E2 ->b=c%2F2
x=-2c%2B%28%28c%2F2%29%5E2%2Bcy-y%5E2%29-%28c%2F2%29%5E2
x%2B2c%2B%28c%2F2%29%5E2+=%28c%2F2-y%29%5E2
x%2B2c%2Bc%5E2%2F4+=%28c%2F2-y%29%5E2
sqrt%28x%2B2c%2Bc%5E2%2F4%29+=c%2F2-y
y=+c%2F2-sqrt%28x%2B2c%2Bc%5E2%2F4%29
f%5E-1=+c%2F2-sqrt%28x%2B2c%2Bc%5E2%2F4%29

if f%5E-1%283%29=-1, means if x=3 then y=-1
-1=+c%2F2-sqrt%283%2B2c%2Bc%5E2%2F4%29
sqrt%283%2B2c%2Bc%5E2%2F4%29=+c%2F2%2B1....square both sides
3%2B2c%2Bc%5E2%2F4=+%28c%2F2%2B1%29%5E2
3%2B2c%2Bc%5E2%2F4=+c%5E2%2F4%2B2%28c%2F2%29%2B1....simplify
3%2B2c=+c%2B1
2c-c=1-3
c=-2
so, f%28x%29=-2c%2Bcx-x%5E2 is
f%28x%29=-2%28-2%29%2B%28-2%29x-x%5E2
f%28x%29=4-2x-x%5E2

and
f%5E-1%28x%29+will be
f%5E-1=+-2%2F2-sqrt%28x-4%2B1%29
f%5E-1=+-1-sqrt%28x-3%29+-> check result if x=3, f%5E-1=+-1
-1=+-1-sqrt%283-3%29
-1=+-1-sqrt%280%29
-1=+-1->true

Answer by ikleyn(52852) About Me  (Show Source):
You can put this solution on YOUR website!
.


The solution by @MathLover1 is incorrect.

It has an error.

The error is in the 8-th line of her post.



Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Given the function f(x)=-2c+cx-x² and f-1(3)=-1, find c
f-1(3)=-1 simply means that f(-1) = 3

f(-1) = -2c+c(-1)-(-1)²

    3 = -2c-c-1
    3 = -3c-1
   3c = -4
    c = -4/3

Edwin