/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 The functions \(f, g,\) and h ar... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 39

The functions \(f, g,\) and h are defined as follows: $$ f(x)=2 x-3 \quad g(x)=x^{2}+4 x+1 \quad h(x)=1-2 x^{2} $$ In each exercise, classify the function as linear, quadratic, or neither. $$g \circ h$$

Short Answer

Expert verified
The function \( g \circ h \) is neither linear nor quadratic.

Step by step solution

01

Understand the Composition

The composition \( g \circ h \) means \( g(h(x)) \). First, evaluate the function \( h \), then substitute the result into \( g \).
02

Substitute h(x) into g(x)

Substitute \( h(x) = 1 - 2x^2 \) into \( g(x) \), resulting in \( g(h(x)) = (1 - 2x^2)^2 + 4(1 - 2x^2) + 1 \).
03

Simplify the Expression

Expand \( (1 - 2x^2)^2 \) to get \( 1 - 4x^2 + 4x^4 \). Then multiply and simplify \( 4(1 - 2x^2) \) to get \( 4 - 8x^2 \), and add all terms: \( g(h(x)) = 4x^4 - 12x^2 + 6 \).
04

Classify the Function

The resulting function \( g(h(x)) = 4x^4 - 12x^2 + 6 \) includes a term with \( x^4 \), which is the highest degree term. This function is neither linear nor quadratic since its degree is 4.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Function
A linear function is one of the simplest types of functions you will encounter in algebra. It is a polynomial function of degree one. This means the highest power of the variable (usually denoted as \( x \)) is one. A typical form of a linear function is \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

Linear functions create straight lines when graphed. The constant \( a \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis. These functions are used to model relationships that change at a constant rate.

Key Characteristics:
  • Graph: Straight line
  • Equation form: \( f(x) = ax + b \)
  • Degree: 1
Linear functions are easy to work with and form the foundation for understanding more complex functions.
Quadratic Function
A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. Quadratic functions include the standard form \( g(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero.

The graph of a quadratic function is a parabola. It can open upwards or downwards depending on the sign of \( a \). If \( a \) is positive, the parabola opens upward, and if it is negative, the parabola opens downward.

Key Characteristics:
  • Graph: Parabola
  • Equation form: \( ax^2 + bx + c \)
  • Degree: 2
  • Vertex: The highest or lowest point on the graph
Quadratic functions are frequently used to model scenarios where the rate of change is variable.
Degree of Polynomial
The degree of a polynomial indicates the highest power of the variable in its expression. For example, in the function \( 4x^4 - 12x^2 + 6 \), the degree is 4, as the term with the highest power is \( 4x^4 \).

Understanding the degree is crucial as it tells us about the number of roots or potential solutions a polynomial might have, as well as the overall shape of its graph. Polynomials can be classified based on their degree:
  • Degree 0: Constant function \( (c) \)
  • Degree 1: Linear function \( (ax + b) \)
  • Degree 2: Quadratic function \( (ax^2 + bx + c) \)
Higher-degree polynomials, such as cubic functions (degree 3) or quartic functions (degree 4), can have more complex graphs with multiple turning points and roots.

In the original exercise, the expression \( g(h(x)) = 4x^4 - 12x^2 + 6 \) illustrates a polynomial of degree 4, highlighting that it is neither linear nor quadratic.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

On the same set of axes, graph the four parabolas \(y=x^{2}\) \(2 x^{2}, 3 x^{2},\) and \(8 x^{2} .\) Relative to your graphs, where do you think the graph of \(y=50 x^{2}\) would fit in? After answering, check by adding the graph of \(y=50 x^{2}\) to the picture.

(This exercise refers to Example 8.) Let \(y=\frac{(x-3)(x+2)}{(x+1)(x-2)}\) Verify each of the following approximations. (a) When \(x\) is close to \(-2,\) then \(y \approx-\frac{5}{4}(x+2)\) (b) When \(x\) is close to \(-1,\) then \(y \approx \frac{4 / 3}{x+1}\) (c) When \(x\) is close to \(2,\) then \(y \approx \frac{-4 / 3}{x-2}\)

Graph the functions. Notice in each case that the numerator and denominator contain at least one common factor. Thus you can simplify each quotient; but don't lose track of the domain of the function as it was initially defined. (a) \(y=\frac{x+2}{x+2}\) (b) \(y=\frac{x^{2}-4}{x-2}\) (c) \(y=\frac{x-1}{(x-1)(x-2)}\)

Sketch the graph of each rational function. Specify the intercepts and the asymptotes. $$y=\left(4 x^{2}+x-5\right) /\left(2 x^{2}-3 x-5\right)$$

(a) Complete the following table. Which \(x\) -y pair in the table yiclds the smallest sum \(x+y ?\) $$\begin{array}{llllllll} \hline x & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 \\ y & & & & & & \\ x y & 12 & 12 & 12 & 12 & 12 & 12 & 12 \\ x+y & & & & & & & \\ \hline \end{array}$$ (b) Find two positive numbers with a product of 12 and as small a sum as possible. Hint: The quantity that you need to minimize is \(x+(12 / x),\) where \(x>0 .\) But $$x+\frac{12}{x}=(\sqrt{x}-\sqrt{\frac{12}{x}})^{2}+2 \sqrt{12}$$ This last expression is minimized when the quantity within parentheses is zero. Why? (c) Use a calculator to verify that the two numbers obtained in part (b) produce a sum that is smaller than any of the sums obtained in part (a).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.