Intersection points, part 1

Algebra Level 3

$f(x)=x^3-6x^2+14x-12$

Given that the graphs of $f$ and its inverse $f^{-1}$ intersect at $(a,b)$ .

Find the value of $a^2+b^2$ .

The answer is 18.

1 solution

Amzad Hosein
Jul 27, 2016

Since inverse functions are reflected across the line y=x, each invariant point (x,y) is of the form (x,x).

So we only need to find the solutions to x = f(x)

.'. x = x^3-6x^2+14x-12 --> 0 = x^3-6x^2+13x-12

Let g(x) = x^3-6x^2+13x-12 Using the factor theorem, g(3)=0 implies that (x-3) is a factor.

We can prove this is the only real solution by factoring and noticing the quadratic factor has non-real roots, but it can also be deduced by the implications of the question that the point of intersection is unique.

Thus, (3,3) is the point of intersection. And 3^2+3^2 = 18.

Intersection points, part 1

The answer is 18.

1 solution

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