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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 35
Capitolo 1, Problema 35

Composite functions and notation
Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).
Simplify or evaluate the following expressions.
F(g(y))

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1
Identify the functions involved: \( F(x) = \frac{1}{x-3} \) and \( g(x) = x^3 \).
Understand the composition of functions: \( F(g(y)) \) means substituting \( g(y) \) into \( F(x) \).
Calculate \( g(y) \): Since \( g(x) = x^3 \), then \( g(y) = y^3 \).
Substitute \( g(y) = y^3 \) into \( F(x) \): Replace \( x \) in \( F(x) = \frac{1}{x-3} \) with \( y^3 \).
Simplify the expression: \( F(g(y)) = \frac{1}{y^3 - 3} \).

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Composite Functions

A composite function is formed when one function is applied to the result of another function. In this case, F(g(y)) means we first evaluate g(y) and then use that result as the input for F. Understanding how to combine functions is crucial for simplifying expressions involving multiple functions.
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Function Notation

Function notation is a way to denote functions and their inputs clearly. For example, f(x) indicates that f is a function of x. This notation helps in identifying which function to apply and in what order, especially when dealing with composite functions like F(g(y)).
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Multiplying & Dividing Functions

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to understand or compute. In the context of composite functions, this may include substituting values and performing algebraic operations to combine the functions into a single expression, which is essential for evaluating F(g(y)).
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Simplifying Trig Expressions
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Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3). Simplify or evaluate the following expressions.

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Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

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Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

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Draining a tank (Torricelli’s law) A cylindrical tank with a cross-sectional area of 1010 m2 is filled to a depth of 2525 m with water. At t=0t=0 s, a drain in the bottom of the tank with an area of 11 m²  is opened, allowing water to flow out of the tank. The depth of water in the tank (in meters) at time t≥0t\(\geq{0}\) is d(t)=(5−0.22t)2d\(\left\)(t\(\right\))=\(\left\)(5-0.22t\(\right\))^2.

c. What is an appropriate domain for dd?

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