Let f(x)=x-3, g(x)=√(x), h(x)=x^3, and j(x)=2 x . Express each of the functions in Exercises

Let f(x)=x-3, g(x)=√(x), h(x)=x^3, and j(x)=2 x ....

Key Concepts

Function Composition

Function composition is the process of applying one function to the result of another. In problems involving composite functions, an inner function is applied first, and its output is then used as the input to an outer function. This concept is fundamental when rewriting complex expressions as combinations of simpler functions.

Decomposition of Functions

Decomposing a function involves breaking it down into simpler constituent functions whose compositions reconstruct the original function. This technique helps simplify the analysis, manipulation, and computation of functions by leveraging known functions or operations.

Mapping Components to Given Functions

When provided with specific functions, it is important to identify parts of the target function that correspond to those given functions. This often involves recognizing operations such as shifts, scalings, or exponentiations within the target function and matching them to one or more of the given functions.

Order of Operations in Composites

In any composite function, the order in which functions are applied is critical. Understanding that compositions are evaluated from the innermost function outward ensures that the correct sequence of transformations is used to reconstruct the original expression.

Transcript

00:02 Hello, in this problem we're given the definitions of four functions of fgh and j and we need to express the following functions as a composition of some of these functions.

00:23 So the first function is the square root of x minus 3.

00:29 The square root of x minus 3 is something minus 3 which is that if that something is the square root we could say that we could recognize this is f of the square root of x because what f does it takes something and then transforms it to something minus 3 if that something is x then if the argument is x then minus 3 results if the argument is z it would be z minus 3 if the argument is square root of x it's square of x minus 3 so f of the square root of x is the given function now the square of x we recognize as g of x so this is f of g of x if of x if of g of x if of of x is the composition of the functions of f and g the inner function acts first so this would be g of x composite with the function is okay they function is is y is y is equal to two the square roots of x listening through the functions we see that j of x is two times something two times x if we replace x here with square of x we will recognize this as j of the square of x what does j do to the square of x it it transforms let me write it like i did in here so if you take an argument the argument's function value would be two times that argument so if you place square root of x here, two times square of x would be j of square of x.

02:47 Again, square of x is the function g.

02:50 So this is j of g of x, and this is the composition of j and g applied on the function x.

03:01 C function is y is x2, the one quix.

03:10 Quarter.

03:11 Now x to the one quarter is the fourth root of x to the one applying the a the rational exponent.

03:28 When this rational exponent this represents radical, the denominator determines what the index of the root is and the numerator tells us the exponent of the power underneath the root sign.

03:47 So fourth root of x is the same as the second root of x of the second root of x.

03:59 So recognize the second root of x as g x, the square root of x is g, so it's the square root of g, so it's the square root of x.

04:12 And taking a square root, so if you place anything instead of x, g is going to transform it, g is going to transform it into the square root of that value.

04:30 So what this is, this is g of g of x.

04:36 We've taken a square root of g of x, and we write this as g in composition with itself.

04:44 Of x part d y equals 4x y equals 4x we can rewrite this as 2 times 2x now have a look at the inner function we recognize 2x as being the function of j of x so this is 2 times j of x now the function j of x is such that if if you place j of x here then you're going to have two times j of x.

05:23 So this is j of, j of x, which is written as the composition of j with itself of the function x.

05:35 E.

05:36 Part is the square root.

05:39 Y is the square root of x minus 3 squared.

05:45 X minus 3 squared...

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