Find a domain on which f is one-to-one and a formula for the inverse of f restricted to this dom

Find a domain on which f is one-to-one and a formula for the...

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Key Concepts

One-to-One Function

A function is one-to-one if each output value is produced by exactly one input value. This property, also known as injectivity, is fundamental when considering the inverse of a function, as only one-to-one functions have inverses that are also functions.

Inverse Function

The inverse function reverses the roles of inputs and outputs of the original function. Finding an inverse involves solving for the original variable in terms of the function value so that the new function maps each output back to its original input.

Domain Restriction

Restricting the domain of a function is often necessary when the original function is not one-to-one over its entire domain. By limiting the inputs to an interval where the function is monotonic (either strictly increasing or strictly decreasing), the function becomes one-to-one and an inverse can be properly defined.

Graphical Interpretation of Functions and Their Inverses

Graphing a function along with its inverse involves understanding that the inverse function is the reflection of the original function about the line y = x. This visual symmetry helps in confirming that the functions are indeed inverses of each other and provides insight into their respective domains and ranges.

Transcript

00:01 We're going to graph this original function f, which is defined by 1 over root x squared plus 1.

00:08 So we'll just sub some numbers in and see what happens.

00:11 If i put 0 into x, it'll be 1 over 1.

00:14 So that's 1.

00:15 If i put 1 into x, it'll be 1 over root 2.

00:19 So it'll be somewhere lower than 1.

00:22 And negative 1 would give the same result since it's going to square and turn into a positive anyway.

00:28 So we know that the left side and the right side will look the same.

00:33 As i put bigger numbers in, so let's say x equals 2, it'll be 1 over root 5, so that's going to be even smaller.

00:40 And same thing on the other side.

00:41 So we should get a curve that is bounded by an asymptote at zero, horizontal asymptote at zero.

00:49 And it curves up and then comes back down like that.

00:53 Draw our horizontal asymptote in here.

00:58 There we go.

00:59 And our job is to find the inverse, so let's write this as y equals 1 over root x squared plus 1, swap x and y, so 1 over root y squared plus 1, and this will be root y squared plus 1 equals 1 over x, square both sides, y squared plus 1 equals 1 over x, square both sides, equals 1 over x squared minus 1 and y is equal to square root of 1 over x squared minus 1.

01:36 Of course there's a plus minus here...

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