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

step 1 and we're going to calculate the context. We have f of x equals 1 divided by x plus 3. We have g

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

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.
$$
f(x)=\frac{1}{x+3} ; \quad g(x)=-\frac{2}{x}
$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Order of Composition

The order in which functions are composed is important because f ? g generally does not equal g ? f. Each composite function must be computed by following the correct sequence, as each function’s rules and potential restrictions must be applied in the precise order indicated by the composition.

Domain of Composite Functions

The domain of a composite function consists of all values of x that not only lie in the domain of the inner function but also ensure that the corresponding outputs are within the domain of the outer function. This means carefully considering and combining the restrictions from both functions, such as ensuring denominators are nonzero or other constraints that their definitions impose.

Composite Functions

Composite functions involve applying one function to the result of another function. In mathematical notation, if you have two functions f and g, then the composite function f ? g is defined by (f ? g)(x) = f(g(x)). This process requires first computing g(x) and then using this value as the input in function f.

Transcript

00:02 And we're going to calculate the context.

00:06 We have f of x equals 1 divided by x plus 3.

00:10 We have g of x equals minus 2 divided by x.

00:15 The first thing we're going to calculate is f composed with g of x, which equals of g of x.

00:28 And so what this means is we substitute g into f.

00:35 Now this gives 1 over minus 2 divided by x we're replacing this x with g now it's a simplified as a third denominator and that becomes three x minus two divided by x that's just adding fractions together so we'll see that we're taking the reciprocal that's going to be x number three x minus two now we have to look at this denominator to see so when x is equal to two -thirds a division by zero is going to occur.

01:15 So the domain of this function is the real number but without the number two thirds.

01:23 So this is the only number that causes any issues.

01:26 Next we're looking at g composed with f is x and this is equal to g of f as x and so we substitute f into g.

01:38 We have minus two divided we're going to place x but f and that's 1 over x plus 3.

01:45 So we can see that this becomes minus 2 times x plus 3, which we can expand to minus 2x minus 6.

01:55 We'll see that there's no issues with this function, and so it's the main, is the real number.

02:03 Next we're looking at f composed of x, so this is just f of f of x...

Please give Ace some feedback