Textbook Question
Composition of polynomials
Let ƒ be an nth-degree polynomial and let g be an mth-degree polynomial.
What is the degree of the following polynomials?
ƒ ⋅ f
0. Functions
Combining Functions
2:28 minutesProblem 19c
Textbook Question
Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find ƒ(g(h( x))).
Verified step by step guidance1
Identify the innermost function in the composite function \( f(g(h(x))) \). Here, it is \( h(x) = \sqrt{x} \).
Substitute \( h(x) \) into \( g(x) \), resulting in \( g(h(x)) = g(\sqrt{x}) = \sin(\sqrt{x}) \).
Now, substitute \( g(h(x)) \) into \( f(x) \), resulting in \( f(g(h(x))) = f(\sin(\sqrt{x})) \).
Since \( f(x) = x^3 \), replace \( x \) with \( \sin(\sqrt{x}) \) to get \( (\sin(\sqrt{x}))^3 \).
Thus, the expression for \( f(g(h(x))) \) is \( (\sin(\sqrt{x}))^3 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Composite Functions
Composite functions are formed when one function is applied to the result of another function. In mathematical notation, if you have two functions f(x) and g(x), the composite function f(g(x)) means you first apply g to x, and then apply f to the result of g. Understanding how to combine functions is essential for solving problems involving multiple functions.
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Function Notation
Function notation is a way to represent functions and their operations clearly. For example, f(x) denotes a function f evaluated at x. This notation helps in understanding how to manipulate and combine functions, especially when dealing with compositions like f(g(h(x))). Recognizing how to read and interpret these notations is crucial for solving calculus problems.
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Basic Functions
Basic functions such as polynomial functions, trigonometric functions, and root functions form the foundation of calculus. In this question, f(x) = x³ is a polynomial function, g(x) = sin x is a trigonometric function, and h(x) = √x is a root function. Knowing the properties and behaviors of these functions is vital for accurately performing operations like composition.
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