In Exercises 31 to 48 , find f^-1(x). State any restrictions on the domain of f^-1(x). f(x)=x^2-

In Exercises 31 to 48 , find f^-1(x). State any restrictions...

Key Concepts

Inverse Function

An inverse function essentially reverses the effect of the original function. For a function f, its inverse f?¹, if it exists, satisfies the condition that f(f?¹(x)) = x and f?¹(f(x)) = x. Finding an inverse involves solving the equation y = f(x) for x in terms of y, and then interchanging the roles of x and y. It is critical that the function be one-to-one (injective) over its domain to have a well-defined inverse.

Domain and Range Restrictions

The domain of the original function becomes the range of its inverse, and the range of the original function becomes the domain of the inverse. Especially when dealing with functions that are not one-to-one over their entire domain, such as quadratics, the function must be restricted to an interval where it is monotonic. In such cases, the restrictions on the original function are essential so that its inverse is properly defined over a corresponding interval.

Completing the Square

Completing the square is a technique used to rewrite quadratic functions in vertex form, which simplifies the process of inverting them. By rewriting the quadratic function in the form (x - h)² + k, one can more easily solve for x in terms of y. This method is particularly useful when the quadratic has been limited to a specific interval, ensuring that only the appropriate branch of the square root is considered during inversion.

Transcript

00:01 In this problem, the first thing we're being asked to do is to find the inverse of f of x.

00:04 So remember, to do that, we first invert and switch our x's and y values.

00:09 Keep in mind, f of x is equal to y.

00:11 So when we switch x and y, that will give us x equals y squared plus 6y plus 1.

00:19 So now what we're going to do is we're going to solve this for y by using the completing the square method.

00:25 So first, i'll subtract one from both sides.

00:27 So x minus 1 will equal to y squared plus 6y.

00:32 Now we have to find the constant that will make this a perfect square trinomial.

00:36 To do this, we use the formula b over 2 squared, where b is the coefficient of our y term, which is 6.

00:42 So we'll have 6 divided by 2, which is 3, and 3 squared is 9.

00:47 So we're going to add 9 to both sides to keep it balance.

00:51 Well, negative 1 plus 9 is 8.

00:52 So if x plus 8 equals, well remember on the right hand side, we now have a perfect.

00:57 Square trinomial.

00:58 So we'll factor to one binomial being squared.

01:02 The first term of the binomial is the square root of the first term, which is y.

01:06 The second term is the square root of the last term, which is three.

01:10 And we always use the sign in the middle term, which is plus.

01:13 Perfect...

Please give Ace some feedback