m(x) = x² + 4x + 9

Parabola , positive coefficient of x² means U-shaped (upward sloping curve)

Complete the square to find the coordinates of the minimum point.

x² + 4x + 9

= x² + 2(2)x + 2² + 5

= (x + 2)² + 5

= (x - (-2))² + 5


So the minimum point has coordinates (-2,5)


For the function to have an inverse, it must be one-one. Every y-coordinate can only have one x-coordinate corresponding to it.

So x > -2 since the curve is symmetric about the line x = -2.

If x is smaller than -2, then the function will not be one-one since some points on the right side of the curve will have a 'reflection' on the left side.

Least value of a = 2

m(x) = x² + 4x + 9

Replace x with m-¹(x) and m(x) with x.

x = (m-¹(x))² + 4m-¹(x) + 9

x = (m-¹(x) + 2)² + 5

x - 5 = (m-¹(x) + 2)²

±√(x - 5) = m-¹(x) + 2

m-¹(x) = ±√(x - 5) - 2


To determine which one is rejected , first consider the original function with the domain restricted.

The minimum point is (-2,5) and all the y-coordinates are positive.

The inverse is basically a reflection of the curve about the line y = x

The x-coordinates on m will correspond to ('mapped to') y-coordinates on m-¹ and vice versa.

The domain of m is the range of m-¹, vice versa.

So, this tells us that the range of m-¹ is (-2,∞) and its domain is (5,∞)

So, the negative portion is rejected since it would mean that for all real values of x where x > 5, m-¹(x) will be smaller than -2.


Therefore, m-¹(x) = √(x - 5) - 2

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Great explanation