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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 1 - FunctionsProblema 1.2.12f
Capitolo 1, Problema 1.2.12f

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.


f. y = √(x³ − 3)

Guida verificata passo dopo passo
1
Identify the innermost function in the expression y = √(x³ − 3). Here, the innermost operation is x³, which corresponds to the function h(x) = x³.
Next, observe that after applying h(x), the expression becomes h(x) = x³. The next operation is subtraction by 3, which corresponds to the function f(x) = x − 3.
Combine these two operations: first apply h(x) to get x³, then apply f(x) to get x³ − 3. This can be expressed as f(h(x)).
Finally, the outermost operation is taking the square root, which corresponds to the function g(x) = √x. Apply g to the result of f(h(x)) to get g(f(h(x))).
Thus, the composition of functions that represents y = √(x³ − 3) is g(f(h(x))).

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Composition of Functions

Composition of functions involves combining two or more functions to create a new function. If you have 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.
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Function Notation

Function notation is a way to represent functions and their outputs. For example, f(x) represents the output of function f when the input is x. This notation is crucial for understanding how to evaluate functions and perform operations like composition. Recognizing how to read and interpret function notation helps in identifying the relationships between different functions.
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Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations or express functions in different forms. This skill is vital when working with compositions, as it allows you to substitute and combine functions effectively. Mastery of algebraic techniques, such as factoring, expanding, and simplifying, is necessary to express complex functions in terms of simpler ones.
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Copy and complete the following table.


d. <IMAGE>

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