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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 82d
Chapter 1, Problem 82d

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 o g

Verified step by step guidance
1
Identify the degree of the polynomial \( f(x) \), which is \( n \).
Identify the degree of the polynomial \( g(x) \), which is \( m \).
Understand that the composition \( f \circ g \) means substituting \( g(x) \) into \( f(x) \).
The degree of \( f \circ g \) is determined by multiplying the degree of \( f(x) \) by the degree of \( g(x) \).
Therefore, the degree of \( f \circ g \) is \( n \times m \).

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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) = 2x^3 + 3x^2 + 1, the degree is 3. Understanding the degree is crucial for determining the behavior of the polynomial, including its end behavior and the number of roots.
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Introduction to Polynomial Functions

Composition of Functions

Function composition involves combining two functions where the output of one function becomes the input of another. For polynomials, if f(x) and g(x) are two functions, the composition f(g(x)) is evaluated by substituting g(x) into f(x). This process is essential for analyzing the resulting polynomial's degree and behavior.
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Evaluate Composite Functions - Special Cases

Degree of Composed Polynomials

When composing two polynomials, the degree of the resulting polynomial f(g(x)) is determined by multiplying the degrees of the individual polynomials. Specifically, if f is an nth-degree polynomial and g is an mth-degree polynomial, then the degree of the composition f(g(x)) is n * m. This concept is vital for predicting the complexity of the resulting polynomial.
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Related Practice
Textbook Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

ƒ(x)=x5−x3−2ƒ(x)=x{^5}-x^3-2

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Textbook Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

x23+y23=1x^{\(\frac\)23}+y^{\(\frac\)23}=1

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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

Textbook Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

tan(tan−11)\(\tan\]\left\)(\(\tan\)^{-1}1\(\right\))

Textbook Question

Parabola properties Consider the general quadratic function ƒ(x) = ax² + bx + c , with a ≠ 0.


a. Find the coordinates of the vertex of the graph of the parabola y= ƒ(x)  in terms of a, b, and c.

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Textbook Question

A culture of bacteria has a population of 150150 cells when it is first observed. The population doubles every 12 hr12~\(\text{hr}\), which means its population is governed by the function p(t)=150⋅2t12p\(\left\)(t\(\right\))=150\(\cdot{2^{\frac{t}{12}\)}}, where tt is the number of hours after the first observation.

How long does it take the population to reach 10,00010,000?

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