Domanda del libro di testo
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)
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 1, Problema 1.2.16e
Composition of Functions
Evaluate each expression using the functions
f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0
x − 1, 0 ≤ x ≤ 2
e. g(f(0))
Guida verificata passo dopo passo1
First, identify the function f(x) = 2 - x. We need to evaluate f(0) first.
Substitute x = 0 into f(x): f(0) = 2 - 0.
Calculate f(0) to find the result, which will be used as the input for g(x).
Next, use the result from f(0) as the input for g(x). Determine which piece of the piecewise function g(x) to use based on the value of f(0).
Evaluate g(f(0)) by substituting the value of f(0) into the appropriate piece of g(x) and simplify the expression.
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Function Composition
Function composition involves combining two functions to create a new function. If you have two functions, f(x) and g(x), the composition g(f(x)) means you first apply f to x, and then apply g to the result of f. This process is essential for evaluating expressions where one function's output becomes the input for another.
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3:48
Evaluate Composite Functions - Special Cases
Piecewise Functions
A piecewise function is defined by different expressions based on the input value. In the given problem, g(x) is defined differently for two intervals: one for values from -2 to 0 and another for values from 0 to 2. Understanding how to evaluate piecewise functions is crucial for correctly applying them in function composition.
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Percorso guidato
05:36Piecewise Functions
Evaluating Functions
Evaluating a function involves substituting a specific value into the function's expression to find the output. For example, to evaluate f(0) in the function f(x) = 2 - x, you replace x with 0, resulting in f(0) = 2. This step is necessary before performing function composition, as it determines the input for the next function.
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Percorso guidato
4:26Evaluating Composed Functions
Pratica correlata
Domanda del libro di testo
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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Domanda del libro di testo
Composition of Functions
Evaluate each expression using the functions
f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0
x − 1, 0 ≤ x ≤ 2
f. f(g(1/2))
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Domanda del libro di testo
Combining Functions
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?
g. g ∘ f
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Domanda del libro di testo
Combining Functions
Assume that f is an even function, g is an odd function, and both f and g are defined on the entire real line (−∞,∞). Which of the following (where defined) are even? odd?
d. f² = ff
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Domanda del libro di testo
Composition of Functions
Copy and complete the following table.
d. <IMAGE>
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