Find a formula for the inverse of the function. f(x)=2 x^2-8...

Key Concepts

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression into a perfect square form, which simplifies the process of solving for the variable. This method is particularly useful when deriving the inverse of a quadratic function, as it makes isolating the variable more straightforward.

Domain Restriction

Restricting the domain of a function is crucial when finding an inverse for functions that are not naturally one-to-one. For quadratic functions like f(x) = 2x² - 8x, limiting the domain ensures the function is monotonic, thereby enabling the existence of a well-defined inverse.

Square Root Function and Branch Selection

When solving a quadratic equation by taking the square root of both sides, it is important to select the appropriate branch (positive or negative) that corresponds to the restricted domain of the original function. This ensures that the inverse function correctly mirrors the behavior of the original function over its restricted domain.

Inverse Function

An inverse function essentially reverses the operation of the original function, mapping outputs back to their corresponding inputs. For a function f, its inverse f?¹ satisfies f(f?¹(y)) = y and f?¹(f(x)) = x, provided that f is one-to-one on its domain.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, and their graphs are parabolas. Unlike linear functions, quadratic functions are not one-to-one over their entire domain and typically require a restriction on the domain to ensure that an inverse function exists.

Transcript

00:01 All right, we have this parabola here, and it's only defined for x greater than equal to 1, because its domain has to be restricted.

00:12 So this is the domain.

00:14 And remember, domain means what x values you can put into the function.

00:20 Okay, so we need to have a look at this parabola so we can make sure that it's one -to -one so that it actually does have an inverse.

00:28 And because i'm going to need it later, i'm going to go ahead and figure out where it is by completing the square.

00:35 So i'm going to add something and i'm going to subtract something.

00:38 So i'm going to add half of minus two squared.

00:45 So half of minus two is minus one and minus one squared is one.

00:49 Oops.

00:50 So i'm going to add one and at the same time i'm going to subtract one.

00:54 Okay.

00:55 So i haven't changed the equation.

00:57 It's still x squared minus 2x.

00:59 But now i can write it in this form.

01:01 This is a perfect square.

01:02 That's why it's called completing the square.

01:04 It's x minus 1 squared minus 1.

01:08 Okay, now i can see that the vertex of this parabola is 1 minus 1.

01:16 Okay, because when it's in this form, x minus h squared plus k, then hk is the vertex.

01:27 Okay, so there's a minus here.

01:29 That means h is 1, but there's a minus here.

01:32 That means k is minus 1.

01:35 All right.

01:36 So we have a vertex at 1, 1.

01:41 And if we would have just factored this in the first place, it's x times x minus 2, we have x intercepts at 0 and 2.

01:49 Okay, so now we have a pretty clear picture of what the parabola looks like.

01:56 Oh, it's most beautiful parabola.

01:59 Okay, but now we have to restrict it, only use the parabola for the part of the graph where x is greater than or equal to 1.

02:09 So here's one.

02:10 So, here's one.

02:10 We're going to keep this half over here, and we're going to get rid of the rest of it.

02:19 Okay, so we also need to know what is the range of this function...

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