Can We Find The Inverse of A Cubic Function?

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❤️ f(x) = x^3 + 3x, find f^(-1)(x) = ?

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[NadiehFan]

Note that for f(x) = x³ + 3x we have f'(x) = 3x² + 3 which is positive for any real x. So, f(x) is strictly increasing on ℝ and therefore invertible as a real function. Noting that f(0) = 0 and f(x) = x³(1 + 3/x²) for x ≠ 0 we have f(x) > x³ for x > 0 and f(x) < x³ for x < 0 so the range of the function is ℝ since it is a continuous function. Consequently, for any real y the cubic equation x³ + 3x = y has a _single_ and therefore unique real solution in x.

For any real y, the equation x³ + 3x = y also has two conjugate complex solutions which can be expressed by multiplying each of the cube roots of the real solution by one of the two complex cube roots of unity in either order. Obviously, f(x) = x³ + 3x is therefore not invertible when its domain is extended to the set ℂ of all complex numbers.

[cosmosapien]

There's a formula for finding the roots of a cubic equation. So you can find x in terms of y.

[fullgasinneutral]

Of course, the two other solutions for x come from the interchangeable symmetry of a and b.
Let's call the two possible values for a according to the sign in front of the square root (time stamp 5:58): a+ and a-. Due to the symmetry between a and b we can have the following four pairs: (1) a = a+, b=a- (the pair you chose), (2) a=a+, b=a+, (3) a=a-, b=a- and (4) a=a-, b=a+.
However since x =a+b, pair (1) and pair (4) give you the same solution for x, so we get only three distinct inverse functions.

[bartolhrg]

4:30 I tend to use capital letters for such cases A=a³
And I would also introduce it earlier at around 3:50

[hafizusamabhutta]

How do you make these videos? Which app do you use? Which device?

[ingenuintrutschnig]

Shouldn't it be
x³ + 3 x, not
x³ + x...