For the given functions f and g, find: (a) f ∘g (b) g ∘f (c) f ∘f (d) g ∘g State the domain of

For the given functions f and g, find: (a) f ∘g (b) g ∘f...

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=2 x+3 ; \quad g(x)=3 x$$

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Key Concepts

Function Composition

Function composition is the process of applying one function to the results of another, typically written as (f ? g)(x) = f(g(x)). This operation creates a new function where the output of the inner function becomes the input for the outer function. It is a fundamental concept in mathematics that enables complex operations to be built from simpler functions.

Domain of Composite Functions

The domain of a composite function is determined by considering the domains of both the inner and outer functions. For a composite function (f ? g)(x), one must first ensure that x is in the domain of g, and then verify that g(x) falls within the domain of f. This systematic approach ensures that the composite function is well-defined for the chosen inputs.

Transcript

00:01 Okay, so our functions here are f of x equals 2x plus 3, and g of x is equal to 3x.

00:08 So first we're going to find here, f composed g of x, so f composed g of x means f of g of x, where we take g of x is our inside function and we input it into f.

00:21 So f of g of x is equal to, well, f of 3x.

00:26 That's equal to 2 times 3x and then plus 3, which equals, it's equal to 6x plus 3.

00:37 So the, so f composed g of x is equal to 6x plus 3.

00:43 And we're asking to state the domain here.

00:46 We can see this is just a linear function.

00:49 And the domain is going to be the set of all real numbers.

00:53 So we can say r or negative infinity to infinity, right? the set of all real numbers.

01:00 Okay.

01:01 And then for b, we have g -compose f of x.

01:07 That's going to be g of f of x.

01:12 So our inside function is now f.

01:14 We input it into g.