Let P be a polynomial of degree n (a) Can P have an inverse...

Key Concepts

Polynomial Degree and End Behavior

The degree of a polynomial affects its end behavior, which in turn can influence its injectivity. Even-degree polynomials tend to have similar behavior at both ends of the real line, often creating a scenario where the function is not one-to-one, while odd-degree polynomials have opposite end behaviors that can allow for monotonic sections and the possibility of being injection overall.

Monotonicity

Monotonicity is crucial for ensuring injectivity, as a strictly monotonic function (strictly increasing or strictly decreasing) never takes the same value twice. Polynomials that are strictly monotonic over ? are therefore invertible, while those that are not cannot have an inverse on ?.

Local Extrema in Polynomials

Local extrema (peaks and valleys) in polynomials indicate where the function changes direction. For a polynomial to be injective, it must not have multiple local extrema that cause the function to repeat output values, as seen in many higher-degree or non-monotonic odd-degree polynomials.

Inverse Function

An inverse function is defined as a function that 'reverses' the effect of the original function. For a function to have an inverse, it must be bijective (one-to-one and onto), meaning each output is generated by exactly one input.

Injectivity

For a function to be invertible, it must be injective (or one-to-one), which means that no two different elements in the domain map to the same element in the codomain. In the context of polynomials, this property must hold over the entire set of real numbers for the inverse to exist.

Transcript

00:01 All right, for this problem, we are looking at sort of the theoretical properties of polynomials.

00:09 So for the first part of the problem, we want to answer the question, can a polynomial of even degree have an inverse? so specifically, this would be, can it have a proper inverse function? you know, if we have something that has multiple y values corresponding to a single x value, like, you know, if we have something that does that, that's not properly a function.

00:35 So if that is the inverse of something, it would be the inverse of something along those lines, then that wouldn't count as a function.

00:43 So you'd have to say that whatever the original thing was, it doesn't have an inverse.

00:47 So one thing that hopefully should be familiar from the work that you've done on understanding the behavior of polynomials would be that if n is even, then you would expect to have either continuing, you know, coming in from positive at negative infinity and going off to positive at positive infinity, or, you know, something roughly like that, or coming in from negative infinity and going off to positive infinity.

01:28 Sorry, going off to negative infinity as x goes towards positive.

01:32 For example, we can think of x squared, and you know, x squared can pass or a prabla can pass through the origin or through the x -axis multiple times, or it could not pass through it at all.

01:49 And as you get into higher order polynomials, you start getting the sort of flat region towards the minimum there.

01:59 But the key thing is that for any of those shapes, necessarily if we come down from positive infinity and then go back up towards positive infinity, then necessarily you will always have to have some points where you cross through the same y values again, which means that when you go over to the inverse, then it will always be multivalued.

02:38 You'll always, it will always fail.

02:42 The vertical line test, which means that a polynomial, so we can say no, a polynomial of even degree will never have an inverse.

03:10 Alternatively, you can think about this.

03:11 One way that i've seen phrased is that a polynomial of even degree will always have at least one turning point.

03:20 And so, because of that, if there is a turning point where it turns around and starts going back the way that it came in terms of y values, then that means that you'll always have to pass through a value that you already went through.

03:34 So you can't have a proper inverse.

03:36 For the second part of the problem, we want to answer, can a polynomial of odd degree have an inverse? so odd degree polynomials, you know, look along the lines of the straight line or the third degree polynomial.

03:51 And so on, but they always, whatever behavior they have as they go towards negative infinity will have to be the opposite as they go towards positive infinity...

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