Textbook Question
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
a. <IMAGE>
Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.2.13b
Composition of Functions
Copy and complete the following table.
b. <IMAGE>
Verified step by step guidance1
Identify the functions involved in the composition. Let's say we have two functions, f(x) and g(x). The composition of these functions is denoted as (f ∘ g)(x), which means f(g(x)).
Determine the domain of the inner function, g(x). This is the set of all x-values for which g(x) is defined.
Evaluate the inner function, g(x), for the given x-values in the table. Substitute each x-value into g(x) to find the corresponding output.
Substitute the output of g(x) into the outer function, f(x). For each x-value, use the result from g(x) as the input for f(x) to find the final output.
Complete the table by filling in the results of the composition (f ∘ g)(x) for each x-value. Ensure that each step follows logically from the previous one, and check for any restrictions in the domain of f(x) that might affect the composition.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Composition of Functions
The composition of functions involves combining two 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 first apply g to x, and then apply f to the result of g. Understanding this concept is crucial for solving problems that require evaluating or manipulating functions in calculus.
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Function Notation
Function notation is a way to represent functions and their operations clearly. It typically uses symbols like f(x) to denote a function f evaluated at x. This notation helps in understanding how to manipulate and combine functions, especially when dealing with compositions, inverses, or transformations. Familiarity with function notation is essential for interpreting and completing function-related tasks.
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Domain and Range
The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). When composing functions, it is important to consider the domain of the inner function and how it affects the overall composition. Understanding domain and range ensures that the composed function is valid and helps avoid undefined expressions.
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Related Practice
Textbook Question
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
b. <IMAGE>
Textbook Question
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.
a. y = 2x − 3
1
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Textbook Question
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?
b. f/g
Textbook Question
Piecewise-Defined Functions
Find a formula for each function graphed in Exercises 29–32.
b. <IMAGE>
Textbook Question
Theory and Examples
The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.
a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)
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