Question

Find the inverse of this function:

f(x) = x^2 + 2x - 1, x>0

So far I just switched the x and y's...so it looks like this now:

x = y^2 + 2y - 1

I know you're solving for y, and I want to factor (or use quadratic formula)...but x is in the way. So I'm not sure what to do.

Thank you

Answer 1

You will have to treat the x as part of the c term in ax^2 + bx + c = 0

so you have

y^2 + 2y - 1 - x = 0

so a = 1 , by = 2, and c = -1-x

y = (-2 ± √(4 - 4(1)(-1-x)))/2
= (-2 ± √(8+4x))/2
= (-2 ± 2√(2+x))/2
= -1 ± √(2+x)

check with some value of x

e.g. f(5) = 25 + 10 - 1 = 34

so put that in the inverse,
y = -1 ± √(2+34)
= -1 ± √36
= -1 ± 6 = 5 or -7

notice the new inverse is clearly not a function, but we did get our 5 back.