In Exercises 27-32, confirm that f and g are inverses by showing that f(g(x))=x and g(f(x))=x .

step 1 Hey, what's up guys? It's Eric. And today we're going to be confirming if F and G of X are inver

In Exercises 27-32, confirm that f and g are inverses by...

Key Concepts

Composite Functions

Composite functions involve applying one function to the result of another function. In the context of inverse functions, demonstrating that the composition of a function and its alleged inverse yields the identity function (i.e., f(g(x)) = x and g(f(x)) = x) is a standard method to confirm that the two functions are indeed inverses. This relies on the principle that composition of functions is associative and that the order of composition matters.

Inverse Functions

An inverse function reverses the effect of the original function. For a function f, its inverse f?¹ satisfies the condition that f(f?¹(x)) = x and f?¹(f(x)) = x for every x in the domain. This means if you first apply f and then its inverse, you get back to your original input.

Transcript

00:01 Hey, what's up guys? it's eric.

00:02 And today we're going to be confirming if f and g of x are inverses by showing if f of g of x equal to x and if g of f of x is also equal to x.

00:14 So in this problem, we're given that f is equal to x plus 3 over 4, and g of x is equal to 4x minus 3.

00:21 So in order to do that, we're just going to do a heck of a ton of substituting, simplifying, and solving.

00:27 So here we're going to be testing f of g of x minus 3.

00:30 First and testing if that's equal to x so here i've written it as a f of g of x is equal to f of substituting in g of x which is 4x minus 3 into the f of x into x 4 f of x and substituting that whole expression into the function for f of x which is x plus 3 over 4 that turns into 4x minus 3 plus 3 over 4 and i see the same skeleton as f of x over here and we're going to be testing if this is equal to x at all.

01:04 So after this, we're going to be simplifying and solving.

01:06 So from here, we see that 4x minus 3 plus 3 cancels out the minus 3 because of minus 3 plus 3.

01:15 After that, we left with 4x divided by 4.

01:18 And 4x divided by 4 clearly is x.

01:21 So f of g of x does indeed equal x.

01:25 And so that's the first half of our confirmation tests.

01:28 The next half is testing if g of f of x is equal to x.

01:34 So i've gone ahead and i've written that out over here...