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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 82d
Capitolo 1, Problema 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

Guida verificata passo dopo passo
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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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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Evaluating Composed Functions
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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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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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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

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

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

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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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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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