Show that the inverse function of y=x^2-2x+5, x≥1 is f^-1(x)=1+√x-4
Help me please i've been stuck on this question for a few hours and i've lost hope. argh.
the inverse function is: do it yourself before asking ok (;
-_- not funny jo
To find the inverse of a function, we need to interchange the roles of x and y and solve for the new y.
1. Start with the given function:
y = x^2 - 2x + 5
2. Interchange x and y:
x = y^2 - 2y + 5
3. Rearrange the equation:
x - 5 = y^2 - 2y
4. Complete the square:
x - 5 + 1 = y^2 - 2y + 1
x - 4 = (y - 1)^2
5. Take the square root of both sides:
√(x - 4) = √((y - 1)^2)
6. Solve for y:
√(x - 4) = y - 1
7. Add 1 to both sides:
1 + √(x - 4) = y
8. Replace y with f^(-1)(x):
f^(-1)(x) = 1 + √(x - 4)
Therefore, the inverse function of y = x^2 - 2x + 5, for x ≥ 1, is f^(-1)(x) = 1 + √(x - 4).