In Problems 23-38, for the given functions f and g, find: (a) f ∘g (b) g ∘f (c) f ∘f (d) g ∘g St

step 1 F of x equals 1 over x plus 3, and G of x equals negative 2 over x. So here, F of x, our domain

In Problems 23-38, for the given functions f and g, find:...

In Problems $23-38,$ for the given functions $f$ and $g,$ find:
$(a) f \circ g$
(b) $g \circ f$
(c) $f \circ f$
(d) $g \circ g$ State the domain of each composite function.
$$
f(x)=\frac{1}{x+3} ; \quad g(x)=-\frac{2}{x}
$$

Key Concepts

Function Composition

Function composition is the process of applying one function to the result of another to create a new function. This involves substituting the output of one function as the input of the other, leading to an expression like f(g(x)) or g(f(x)).

Order of Composition

The order of composition is key because f composed with g (f ? g) is generally different from g composed with f (g ? f). The inner function's output becomes the input for the outer function, and reversing the order can lead to entirely different results.

Domain of Composite Functions

Determining the domain of a composite function requires ensuring that for each input, the value falls within the domain of the inner function, and that the result of the inner function lies within the domain of the outer function. This dual restriction ensures that the composite function is defined at the chosen inputs.

Transcript

00:04 F of x equals 1 over x plus 3, and g of x equals negative 2 over x.

00:23 So here, f of x, our domain such that x cannot equal negative 3.

00:38 That would make the denominator is 0.

00:41 And g of x, x such that x cannot equal zero.

00:55 So now i go on to a, which is f of g of x, f of negative 2 over x.

01:15 So in my f function, i have 1 over negative 2 over x plus 3.

01:28 So that's your composition, and the rest is your algebra.

01:33 So we have one over, and now i'm simplifying this denominator.

01:38 Negative 2 over x.

01:41 My common denominator would be x, so i have 3x over x.

01:50 Simplifying it further, i have 1 over negative 2 plus 3x over x.

01:59 And now i'm going to invert and multiply.

02:02 So my answer is going to be x.

02:06 Over negative 2 plus 3x.

02:20 And my exclusions from my domain, right here, x cannot be 0, or right here i have to figure out what this cannot be.

02:33 So if negative 2 plus 3x equals 0, 3x would equal 2, divide by 3, x cannot equal two -thirds that would make that a zero so x such that x cannot equal zero x cannot equal two -thirds let's look at b g of f of x so f of x g of one over x plus three so i go to my g function negative 2 over 1 over x plus 3.

03:55 That's your composition.

03:57 Now we do our simplification.

04:01 So i'm just going to flip this and multiply it by my 2.

04:05 So i am 2 over 1 times x plus 3 over 1...

Please give Ace some feedback