In Problems 33-42, verify that the functions f and g are...

In Problems $33-42$, verify that the functions $f$ and $g$ are inverses of each other by showing that $f(g(x))=x$ and $g(f(x))=x$. Give any values of $x$ that need to be excluded from the domain of fand the domain of $g .$
$$f(x)=3-2 x ; \quad g(x)=-\frac{1}{2}(x-3)$$

Key Concepts

Inverse Functions

Inverse functions are functions that 'undo' each other. If f and g are inverses, then applying f followed by g (or vice versa) returns the original input. This relationship is expressed as f(g(x)) = x and g(f(x)) = x. Inverse functions are fundamental in algebra because they allow for the reversal of operations, providing solutions to equations and supporting transformations in various areas of mathematics.

Function Composition

Function composition involves applying one function to the result of another function. When verifying that two functions are inverses, their compositions must yield the identity function, meaning that the output is exactly the input. This process is crucial in confirming the inverse relationship, and it also helps in understanding how different functions interact when combined.

Domain Restrictions

Domain restrictions refer to the values that are excluded from the set of possible input values of a function to ensure that the function remains valid and does not produce undefined expressions. When dealing with inverse functions, it is important to consider domain restrictions because the domain of one function becomes the range of its inverse. Proper identification of these restrictions is crucial in accurately describing the functions and avoiding issues like division by zero or taking square roots of negative numbers.

Transcript

00:01 In this exercise, two functions are provided, and we want to check if they are inverses to each other.

00:07 The first step is going to be to evaluate f of g at x, and the second step will be to reverse the composition to evaluate g at f at x.

00:17 Let's start with the first composition.

00:19 By definition, this is f at, g at x.

00:24 But we know that g of x is defined as follows, so let's make a substitution here and write in negative 1 half times x minus 3 in place of the function g at x so we copy this in by substitution now we use the definition of what f of x does f is the function that takes 3 minus 2 times whatever our input happened to be but we're throwing in g at at our input which by a prior substitution is negative 1 half times x minus 3 so now a rate to simplify this result, this is now 3, and if we distribute a negative 2, or just multiply negative 2 by negative 1 half, we'll get plus 1 times the group x minus 3, which means we can drop the parentheses and say this is 3 plus x minus 3, which is altogether equal to x.

01:24 Since we got an x here, we know that these functions might be inverses to each other, but we have to go to the second step and obtain x.

01:33 If we can get to that point, then we know that they are definitely inverses.

01:38 So let's evaluate to the second composition.

01:41 By definition, it is g at, f at x.

01:45 Then we follow our same strategy.

01:48 We're on the next step.

01:49 I'll write that this is equal to g at, but in place of where f of x was, use the definition that it's equal to 3 minus 2x.

01:58 So that's our first substitution...