Find the inverse function of f informally. Verify that f(f^-1(x))=x and f^-1(f(x))=x f(x)=(x-1)

step 1 So let's talk about inverse functions a bit. So let's say there is a function, and the function

Find the inverse function of f informally. Verify that...

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Key Concepts

Function Composition Verification

After determining an inverse function, it is important to verify its correctness through function composition. This involves composing the original function with its inverse in both orders (f(f?¹(x)) and f?¹(f(x))) and showing that the result in each case is the identity function, i.e., it returns the input unchanged. This step confirms that the functions are true inverses of each other.

Inverse Function

An inverse function essentially reverses the operation of the original function, mapping outputs back to their corresponding inputs. For a function f, the inverse function f?¹ satisfies the condition that executing first f and then f?¹ returns the original input, meaning f(f?¹(x)) = x, and vice versa, f?¹(f(x)) = x.

Finding the Inverse

Finding the inverse of a function generally involves expressing the original function in the form y = f(x), swapping the roles of x and y, and then solving this new equation for y. This process yields the inverse function f?¹, which provides the input corresponding to any given output of f.

Transcript

00:02 So let's talk about inverse functions a bit.

00:05 So let's say there is a function, and the function would be, let's do f of x equals x minus 1 over 4.

00:15 And they ask you to find the inverse of this function.

00:18 So what i want to talk about is what's happening on the right hand side of this problem.

00:23 So by order of operations, if you think about if somebody gave you an x to plug in for this x right here, the first thing you would do with that x is you would subtract one.

00:38 After you subtract one, the second thing you would do is divide by four.

00:45 An inverse function will take these two steps and undo them in reverse order.

00:51 So for the inverse, you would go to this green guy, and your first step in the inverse would be the opposite of what's in green here.

01:01 So instead of dividing by four, we would multiply by four.

01:06 And then the second step will be the opposite of the subtracting 1, which would be add 1.

01:12 So if i'm going to write a function that has these two things, and i want an x, and i will call it f inverse, i would multiply by 4, and then add 1.

01:25 And that's your inverse function.

01:28 And to show that it's the inverse, we can always take the function in its inverse and make the composition.

01:35 So let's replace the inside function here.

01:38 So f inverse is right here...

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