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Chapter 9: Problem 63

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following $$ (f g)(x) $$

Short Answer

Expert verified
(fg)(x) = 2x^3 - 18x.

Step by step solution

01

- Understand the Functions

Identify the given functions: \(f(x) = x^2 - 9\), \(g(x) = 2x\), and \(h(x) = x - 3\).
02

- Know the Definition of Function Multiplication

Recall that if \(f\) and \(g\) are functions, then \((fg)(x)\) represents the multiplication of \(f(x)\) and \(g(x)\). This means \( (fg)(x) = f(x) \cdot g(x) \).
03

- Substitute the Functions

Substitute the functions \(f(x)\) and \(g(x)\) into \((fg)(x)\). This gives: \[(fg)(x) = (x^2 - 9) \cdot (2x)\]
04

- Simplify the Expression

Distribute \(2x\) through \(x^2 - 9\) to simplify: \[ (fg)(x) = 2x \cdot x^2 - 2x \cdot 9 \] Simplify further to get: \[ (fg)(x) = 2x^3 - 18x \]

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Key Concepts

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

Algebraic Functions
Algebraic functions are mathematical expressions that use algebraic operations like addition, subtraction, multiplication, division, and exponentiation. These functions can be as simple as linear equations or as complex as polynomials and rational functions. In this exercise, we see three algebraic functions:

  • \( f(x) = x^2 - 9 \)
  • \( g(x) = 2x \)
  • \( h(x) = x - 3 \)
Understanding algebraic functions helps in performing various operations, including addition, subtraction, and importantly, multiplication, as you'll see in the next section.
Multiplication of Functions
Multiplication of functions means combining two functions into one by multiplying their outputs. If you have two functions, \(f(x)\) and \(g(x)\), the product function is written as \((fg)(x) = f(x) \cdot g(x)\). This exercise involves multiplying two functions:

First, recall the functions:
\( f(x) = x^2 - 9 \)
\( g(x) = 2x \)

Next, substitute them into the multiplication formula:
\( (fg)(x) = (x^2 - 9) \cdot (2x) \)

Lastly, simplify the expression by distributing \( 2x \):
\( (fg)(x) = 2x \cdot x^2 - 2x \cdot 9 \). This gives:
\( (fg)(x) = 2x^3 - 18x \).
Polynomial Expressions
Polynomial expressions are a type of algebraic function represented as the sum of terms, each consisting of a variable raised to a non-negative integer exponent and multiplied by a coefficient. The polynomial we have in this case is: \( f(x) = x^2 - 9 \), where:

  • \( x^2 \) is the quadratic term
  • \( - 9 \) is the constant term
When multiplying polynomials, you apply the distributive property. In the example: \( fg(x) \) is \( 2x(x^2 - 9) \). Distributing \( 2x \) across \( x^2 - 9 \) gives \( 2x^3 - 18x \). Understanding polynomial expressions and how to manipulate them is crucial for solving algebraic function multiplication problems efficiently.

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Most popular questions from this chapter

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (f \circ g)\left(-\frac{1}{2}\right) $$

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ y=\frac{1}{4 x+2} $$

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ y=2 x-6 $$

For each pair of functions, find \(\left(\frac{f}{g}\right)(x)\) and give any \(x\) -values that are not in the domain of the quotient function. $$ f(x)=18 x^{2}-24 x, \quad g(x)=3 x $$

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (h \circ g)(x) $$
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