/*! 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 2 Let \(f(x)=3 x+5\) and \(g(x)=x^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 2

Let \(f(x)=3 x+5\) and \(g(x)=x^{2} .\) Perform each function operation. $$ g(x)-f(x) $$

Short Answer

Expert verified
The simplified expression for \(g(x)-f(x)\) is \(x^{2}-3x-5\).

Step by step solution

01

Write out the functions

The function \(f(x)=3x+5\) and \(g(x)=x^2\). We are asked to compute \(g(x)-f(x)\), which means subtracting the entire \(f(x)\) expression from \(g(x)\).
02

Subtract f(x) from g(x)

To find the difference \(g(x)-f(x)\), replace \(f(x)\) and \(g(x)\) in this expression with their respective function expressions. This gives us \(x^2-(3x+5)\).
03

Simplify the expression

We simplify this expression to \(x^{2}-3x-5\). This is done by distributing the negative sign across the terms \(3x\) and \(5\) in the subtraction \((3x+5)\) and then combining like terms.

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.

Function Operations
When working with algebraic functions, like in the given exercise, understanding how to perform function operations is crucial. Functions are like machines where you input a value and get a specific output. Function operations involve applying basic arithmetic operations, such as addition, subtraction, multiplication, and division, to these functions.
  • In this case, we are dealing with the subtraction of two functions. Consider the functions \(f(x) = 3x + 5\) and \(g(x) = x^2\).
  • The operation \(g(x) - f(x)\) asks us to subtract \(f(x)\) from \(g(x)\).
  • This means we evaluate both functions separately and then subtract the resulting expressions to find a new function that represents their difference.
When performing such operations, keeping track of positive and negative signs, as well as like terms, is important to avoid errors in calculations.
Polynomial Subtraction
Polynomial subtraction is a specific type of function operation that entails subtracting one polynomial from another. Here, polynomials are expressions consisting of variables and coefficients, usually with varying powers. In the example of \(g(x) - f(x)\), we subtract the polynomial \(3x + 5\) from \(x^2\). The process for polynomial subtraction involves a few key steps:
  • First, write out the full expression for both functions involved in the subtraction.
  • Next, distribute the subtraction across the full polynomial, which means applying the negative sign to each term in the polynomial being subtracted.
  • In this exercise, we have \(x^2 - (3x + 5)\), which becomes \(x^2 - 3x - 5\) after distributing the negative sign.
Understanding how to properly apply this operation can help simplify complex problems and is foundational for working with algebraic expressions.
Simplifying Expressions
Once the polynomial subtraction is completed, the next essential step is simplifying the resulting expression. Simplification involves combining like terms and ensuring the expression is in its simplest form for interpretation or further manipulation. Here is how simplification works:
  • Identify like terms: In our example, after subtraction, we have \(x^2 - 3x - 5\). Each term here is distinct, so there are no like terms to combine.
  • Ensure all operations have been completed correctly: Double-checking negative signs and ensuring all terms have been accounted for ensures accuracy.
  • The expression \(x^2 - 3x - 5\) cannot be simplified further as there are no like terms and no arithmetic operations left to perform.
Simplifying expressions allows you to present solutions in their clearest and most usable form, essential for both calculating results and communicating solutions effectively.

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

The graph of \(y=-\sqrt{x}\) is shifted 4 units up and 3 units right. Which equation represents the new graph? A. \(y=-\sqrt{x-4}+3\) B. \(y=-\sqrt{x-3}+4\) C. \(y=-\sqrt{x+3}+4\) D. \(y=-\sqrt{x+4}+3\)

Graph. Find the domain and the range of each function. \(y=-2 \sqrt[3]{x-4}\)

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. Find the domain and the range of each function. \(y=-\sqrt{2(4 x-3)}\)

Compare the domains and ranges of the functions \(f(x)=\sqrt{x-1}\) and \(g(x)=\sqrt{x}-1\)

List all possible rational roots for each equation. Then use the Rational Root Theorem to find each root. $$ 3 x^{3}-5 x^{2}-4 x+4=0 $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure 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.