find the inverse function of f(x) = x^2 - 2x + 5?

Question

My teacher gave me summer work b/c of its an AP course and I can't remember this one. She gave us the answers but we have to show our work...the answer is √(x - 4) + 1. I need to know how to get this answer. Thanks. =]

Answer 1

if y = x²-2x+5 then the inverse is
x=y²-2y+5 solving for x
x= y²-2y+1+4
x=(y-1)² + 4
x-4=(y-1)²
y-1= ±√(x-4)
y=1±√(x-4)

Answer 2

We have f(x) = x^2 -2x +5 ==> y = x^2 -2x + 5. To find the inverse, we change x to y, and y to x and solve for the new y
y = x^2 -2x + 5 becomes x = y^2 -2y + 5
Let's solve for the new y.

first of all, let's rewrite the function in this form
y^2 -2y + 5 = x. Let's move 5 to the right sides by subtracting 5 to both sides

y^2 -2y +5 -5 = x - 5
y^2 -2y = x - 5

we will complete the perfect square. Step to complete the perfect square
1. Divide the coefficient of the middle term by 2

2 . Square the answer you found in step 1

3. Add to both sides of the equation the answer you found in step 2.

Here how it goes. 2 is the coefficient of the middle term.
1. I divide 2 by 2 which is 1.

2. I square 1 which is 1.

3. I add 1 to both sides of the equation

So my equation became y^2 - 2y +1 = x - 5 + 1
I factorize the left side of the equation and i compute the right side of the equation
(y - 1) (y - 1) = x - 4
(y - 1)^2 = x - 4
We look for the square root of both sides
y - 1 = √(x - 4)

Add 1 to both sides
y -1 + 1 = √(x - 4) + 1
y = √(x - 4) + 1

Answer 3

Actually f(x) is non-invertible, as it is not an one-to-one function. But it will be invertible, if we suppose its domain is x>=1.

Swap x and y in
y = x^2 - 2x + 5.

You will get:
x = y^2 - 2y +5

Now solve for y:
x = y^2 - 2y + 5
x = (y - 1)^2 + 4
(y -1)^2 = x - 4
y -1 = √(x - 4)
y = √(x - 4) +1

Answer 4

y = x^2 - 2x + 5
Switch x and y,
x = y^2 - 2y + 5
y^2 - 2y + (5-x) = 0
Use quadratic formula,
y = 1±√(x-4)
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Attention: You have two inverse functions. Which one you pick depends on the domain and the range of the f(x) = x^2 - 2x + 5.

Answer 5

y = x^2 - 2x + 5 = (x - 1)^2 + 4
(x - 1)^2 = y - 4
x - 1 = sqrt(y - 4)
x = 1 + sqrt(y - 4)

g(x) = 1 + sqrt (x - 4) is the inverse of f(x).

Answer 6

Step by step:
f = x^2 - 2x + 5
f - 5 = x^2 - 2x
f - 4 = x^2 -2x + 1
f - 4 = (x - 1) (x - 1)
√ (f - 4) = x -1
√ (f - 4) +1= x

Answer 7

A function must be a bijection (one-to-one and onto) to be invertible. This function is not invertible because it isn't one-to-one.