Textbook Question
Composition of even and odd functions from tables Assume ƒ is an even function, g is an odd function, and both are defined at 0. Use the (incomplete) table to evaluate the given compositions. <IMAGE>
c. ƒ(g(-3))
0. Functions
Combining Functions
1:51 minutesProblem 82c
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
Verified step by step guidance1
Identify the degree of the polynomial \( f \), which is \( n \).
Understand that \( f \cdot f \) represents the composition of the polynomial \( f \) with itself.
Recall that the degree of a polynomial product \( f \cdot g \) is the sum of the degrees of \( f \) and \( g \).
Since \( f \cdot f \) is \( f \) composed with itself, the degree is \( n + n \).
Conclude that the degree of \( f \cdot f \) is \( 2n \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial f(x) = 3x^4 + 2x^2 + 1, the degree is 4. The degree provides important information about the polynomial's behavior, including the number of roots and the end behavior of its graph.
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Multiplication of Polynomials
When multiplying polynomials, the degree of the resulting polynomial is the sum of the degrees of the polynomials being multiplied. For instance, if f is an nth-degree polynomial and g is an mth-degree polynomial, then the degree of the product f ⋅ g is n + m. This principle is crucial for determining the degree of polynomial expressions resulting from operations.
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Composition of Polynomials
The composition of polynomials involves substituting one polynomial into another. For example, if f(x) is a polynomial and g(x) is another, then the composition f(g(x)) results in a new polynomial. While the question focuses on multiplication, understanding composition helps clarify how polynomials interact, especially in more complex expressions.
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