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Chapter 5: Problem 21

Let \(f(x)=-3 x+1\) and \(g(x)=x^{2}+2 .\) Find each of the following. $$ (F+G)(x) $$

Short Answer

Expert verified
(F+G)(x) = x^2 - 3x + 3

Step by step solution

01

Understand the Problem

The problem asks to find \( (F+G)(x) \). This operation involves adding the functions \( f(x) \) and \( g(x) \) together.
02

Write Down the Given Functions

Given \( f(x) = -3x + 1 \) and \( g(x) = x^2 + 2 \).
03

Add the Functions Together

Combine \( f(x) \) and \( g(x) \) by adding their expressions: \[ (F+G)(x) = f(x) + g(x) = (-3x + 1) + (x^2 + 2) \]
04

Simplify the Expression

Combine like terms to simplify the expression: \[ (F+G)(x) = x^2 - 3x + 3 \]

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Key Concepts

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

headline of the respective core concept
To grasp how algebraic functions work, we first need to understand that functions like \( f(x) \) and \( g(x) \) represent rules that assign each input (or \( x \)) a specific output. In this exercise, we have two algebraic functions: \( f(x) = -3x + 1 \) and \( g(x) = x^2 + 2 \). The key task here is to perform the operation of addition on these functions. Algebraic functions can involve various operators and can be linear (like \( f(x) \)) or nonlinear (like \( g(x) \)). When we talk about adding these functions, we are essentially asking what happens to the outputs of these functions when we input the same \( x \) value and add the resulting numbers together.
headline of the respective core concept
Function operations refer to the different ways we can combine two or more functions to produce a new function. In this case, we are focusing on function addition. To add functions \( f(x) \) and \( g(x) \), we simply add their expressions.
  • Write down each function: \( f(x) = -3x + 1 \) and \( g(x) = x^2 + 2 \).
  • Combine these expressions by adding them, which gives us:
\[ (F+G)(x) = f(x) + g(x) = (-3x + 1) + (x^2 + 2) \]This step-by-step process is crucial because each term in the functions must be correctly positioned to ensure proper addition. The combination takes a new form鈥攁 new function that represents the sum of the original functions \( f(x) \) and \( g(x) \).
headline of the respective core concept
Simplification of expressions is the final step in many algebraic operations to make the term easier to work with. After writing out the expression for \( (F+G)(x) \), it鈥檚 important to combine like terms.

For example, starting from our combined expression:

  • \[ (F+G)(x) = (-3x + 1) + (x^2 + 2) \]
  • Identify and combine the like terms (those terms that share the same variable and power):
\[ (F+G)(x) = x^2 - 3x + (1 + 2) \]\[ (F+G)(x) = x^2 - 3x + 3 \]Combining like terms simplifies the function, making it easier to understand and work with in future calculations. This new function \( x^2 - 3x + 3 \) is now the result of adding the original \( f(x) \) and \( g(x) \). Understanding how to simplify expressions is fundamental in algebra, as it allows for more straightforward manipulation and identical expression outcomes.

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Most popular questions from this chapter

Review factoring expressions and solving equations. $$ x+5=0 $$

Simplify. \(\frac{1}{2} x-\frac{1}{4}-\left(\frac{3}{2} x+\frac{3}{4}\right)\)

Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D4 contains the formula $$ =2 * \mathrm{A} 4+3 * \mathrm{B} 4 $$ What is the value in \(\mathrm{D} 4\) if the value in \(\mathrm{A} 4\) is 5 and the value in \(\mathrm{B} 4\) is \(10^{2}?\)

Review factoring expressions and solving equations. $$ 3 x-1=0 $$

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=8 x^{2}-22 x-21, g(x)=2 x-7 $$
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