find the inverse of the function
f(x) = x^2 − 6x, x ≥ 3
Your given function will be the part of the parabola which lies to the right of the vertex (3, -9)
so y = x^2 - 6x
inverse is
x = y^2- 6y
y^2 - 6y = x
complete the square:
y^2 - 6y+ 9 = x+9
(y-3)^2= x+9
y - 3= ± √(x+9)
y = 3 ± √(x+9)
but we only used the part for x≥3 of the original, so
the inverse is
f^-1 (x) = 3 + √(x+9) , x ≥ -9
check:
let x = 4
f(4) = 16 - 24 = -8
f^-1 (-8) = 3 + √(-8+9)
= 3 + √1 = 4
let x = 5.67
f(5.67) = -1.8711
f^-1 (-1.8711) = 3 + √(-1.8711+9)
= 3+√7.1289
=3 + 2.67
= 5.67
I am very confident my inverse is correct!
To find the inverse of a function, we need to follow a few steps:
Step 1: Replace f(x) with y:
y = x^2 - 6x
Step 2: Swap the x and y variables:
x = y^2 - 6y
Step 3: Rearrange the equation to solve for y:
x = y^2 - 6y
Rewrite the equation in the form of a quadratic equation:
y^2 - 6y - x = 0
Step 4: Use the quadratic formula to solve for y:
The quadratic formula is given by: y = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -6, and c = -x.
y = (-(-6) ± √((-6)^2 - 4(1)(-x))) / (2(1))
Simplifying further:
y = (6 ± √(36 + 4x)) / 2
y = (6 ± √(36 + 4x)) / 2
y = (6 ± √(4(9 + x))) / 2
y = (6 ± 2√(9 + x)) / 2
y = 3 ± √(9 + x)
Step 5: Since the given condition is x ≥ 3, we only consider the positive square root to ensure the resulting inverse function is a function.
Therefore, the inverse function is given by:
f^-1(x) = 3 + √(9 + x), x ≥ 3