Composition of Functions

Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.

c. y = x¹/⁴

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Identify the function y = x^(1/4) as a composition of functions. We need to express this function using the given functions f(x), g(x), h(x), and j(x).

Notice that the function y = x^(1/4) can be seen as the fourth root of x, which is equivalent to raising x to the power of 1/4.

Consider the function g(x) = √x, which is the square root of x. We can use this function to help express y = x^(1/4).

To express y = x^(1/4) using g(x), we can rewrite y as g(g(x)). This is because g(g(x)) = √(√x) = x^(1/4).

Thus, the composition of functions that represents y = x^(1/4) is y = g(g(x)).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Composition of Functions

Composition of functions involves combining two or more functions to create a new function. If you have two functions, f(x) and g(x), the composition is denoted as (f ∘ g)(x) = f(g(x)). This means you apply g first and then apply f to the result. Understanding how to manipulate and combine functions is essential for solving problems that require expressing one function in terms of others.

Function Notation

Function notation is a way to represent functions and their operations clearly. For example, f(x) represents a function f evaluated at x. This notation helps in identifying the input and output of functions, making it easier to work with compositions and transformations. Familiarity with function notation is crucial for accurately expressing and manipulating functions in calculus.

Inverse Operations

Inverse operations are processes that reverse the effect of a function. For instance, if g(x) = √x, then g's inverse would be g⁻¹(x) = x², which undoes the square root. Understanding inverse operations is important when composing functions, as it allows for the manipulation of functions to achieve desired forms, such as expressing y = x¹/⁴ in terms of the given functions.

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Related Practice

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The accompanying figure shows the graph of a function f(x) with domain [0,2] and range [0,1]. Find the domains and ranges of the following functions, and sketch their graphs.

<IMAGE>

c. 2f(x)

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Textbook Question

Piecewise-Defined Functions

Find a formula for each function graphed in Exercises 29–32.

b. <IMAGE>

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Textbook Question

Composition of Functions