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 Okay, so here we have f of x is equal to the square root of x minus 2. G of x is equal to 1 minu

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

Key Concepts

Function Composition

Function composition involves applying one function to the result of another. For two functions f and g, the composite function, denoted f ? g, is defined as f(g(x)). This idea is central to understanding how complex functions can be built from simpler ones and is widely used in various mathematical contexts.

Composite Function Domain

The domain of a composite function consists of all x-values in the domain of the inner function for which the output of the inner function lies in the domain of the outer function. This concept is crucial because even if both functions are defined on large sets, restrictions may occur when they are composed.

Radical Functions and Domain Restrictions

Radical functions, especially those involving square roots, require the radicand to be nonnegative in order for the function to be defined in the realm of real numbers. When a composite function involves a radical, it is necessary to determine and apply the constraint that the expression under the square root is at least zero.

Linear Functions

Linear functions, which are typically expressed in the form g(x) = mx + b, have a domain of all real numbers. Their simplicity makes them an ideal candidate in function composition problems because they often do not introduce additional domain restrictions on their own, allowing the focus to be on how their outputs interact with other functions' domain requirements.

Transcript

00:01 Okay, so here we have f of x is equal to the square root of x minus 2.

00:04 G of x is equal to 1 minus 2x.

00:08 So for part a, we're looking to find f composed g.

00:12 That's equal to f of g of x.

00:16 So we start with a function f.

00:18 We input in g.

00:19 That's going to look like, while f of 1 minus 2x is, well, the square root of, while we input in 1 minus 2 x for x.

00:28 We have the square root of 1 minus 2x, then we have minus 2.

00:33 Well, the square root of 1 minus 2x minus 2 is just the square root of, well, negative 2x, 1 minus 2 is minus 1.

00:44 So therefore, this is going to be equal to the square root of negative 2x minus 1.

00:51 Negative 2x minus 1.

00:53 So there is f composed g.

00:56 Now for the domain, well, we have to make sure.

00:58 Sure that g of x is defined, which it always is, right? 1 minus 2x is defined for all x, but we have to make sure that the composite is defined as well.

01:07 So the square root of 1 minus 2x minus 1.

01:10 Well, we have to have the radicand which underneath the square root here has to be greater than equal to 0.

01:15 That implies that x is well less than or equal to negative 1 half, right? because we have a negative sign on front here.

01:25 So therefore we get to here the domain is the set of all x such that x is less than or equal to negative one half all right so there is our domain for f composed g now part b we find g compose f so g compose f means g of f means g of f of x so g of f of x we start with a function g we do g of the square root um of x minus two that's going to look like 1 minus 2 times the square root of x minus 2.

02:07 Okay, and there it is.

02:08 There is g -compose f -of -x, just taking f -x and then inputting it in wherever we see at x in g.

02:16 To get 1 -2 times f -of -x, which is the square root of x minus 2.

02:20 Okay, and then to find our domain, well, we have to have that f -fx is defined where x -fx is defined where x, must be greater than or equal to 2, and that's likewise for g of x.

02:37 So therefore, our domain of g -compose f is the set of all x, such that x is greater than or equal to 2, to make both f of x and the composite function here defined.

02:57 Okay, and then for c, we're finding f -compose f...

Please give Ace some feedback